 I thank the organizers for having given me the opportunity to present these historical ideas in this important colloquium. And as I said, I would like, before I go into substance, to give you some preliminary thoughts I had about preparing this talk. Because as you will see now, it will be quite different from the previous one. And not only because Professor Mark Grange is a great mathematician, I am not. But also because the way in which we prepare now talks nowadays with PowerPoint and so on. So the first thing is to find the pictures of the dramatic person that we are going to speak about. And here we have Dedekin and Frobenius. But these are not the only pictures you find around. There are many others. And hence, I have to think before everything which one I want to pick. Because perhaps if I want to speak about the conflicting views more precisely then I may take these pictures, which are a little bit more conflicting, or this other. But if I want to stress the cooperation, possibly I take this tool to the extent that at some point it is hard to know who is Dedekin and who is Frobenius, as you see here. Hopefully we will not confuse them. So this is the first doubt I had. And I thought about the word Fertauschbar when thinking about these two pictures and probably about the two persons. And this is a very appropriate word here because later on we will speak about Frobenius' article Uwe Fertauschbar in Matrizen. So the idea is Fertauschbarung. It's appropriate over here. Secondly, when you nowadays are going to prepare a talk, sometimes you want to look a little bit in the web. Because perhaps there is some student who wrote a paper on this and you can plagiarize it or something like that. And things are easier. So I looked a little bit about Dedekin, first of all. And of course, if you do Google with Dedekin, you get the standard things, the Dedekin cuts, there's an idea of domains, numbers, and so on. And I decided to take a look at what can I find about cuts. It's not precisely the topic of this talk, but nevertheless. So I find that the Dedekin cut may be something very different from what you have in mind here. It was not of great help for my talk. But I had greater problems with Frobenius because Dedekin, I have written about him. I know better his work. Frobenius is more difficult for me. So I really wanted to find a little bit. And you see here, you have the algebra, theorem in differential geometry, and so on. Some of the names I have heard. I had never heard about the Frobenius substitution. So I decided to click on it. And you have the little summary of what you're going to get and say, Frobenius substitution is the Frobenius most famous find. And moreover, there is, however, no evidence that Frobenius perpetrated the substitution. I said, this is important, perhaps. So let's take a look what is this. Well, this is not about our Frobenius. This is about an ethnographer called Leo Frobenius. My name's Aik, which is nice. And what do we read here? Frobenius' most famous find was the bracelet known as Olokun. He brought it for six pounds and a bottle of whiskey. Underwood, in fact, have demonstrated that the head purporting to be this one, and which is now in Davey Moselle, is not the original, if head, but a copy made by sand casting. And then there is no evidence, however, that Frobenius perpetrated the substitution. The only thing that is not clear here is what a substitution or a permutation, depending on his being here, consistent with his words or not. Well, so I couldn't find a paper by a student, so I had to prepare it by myself. And I have to be a little bit serious, far serious. The thing is that there is a lot of material by historians about this topic. Let me mention some of it, because I did read part of it. I had read it in the past. I now have to reread. Of course, we have this famous and important book by Tom Hawkins on the emergence of the theory of Lee Groot, which is a real masterpiece of the history of mathematics, very demanding, very hard to read, very enlightening, very detailed as well. There is another book by Charles Curtis, a pionist of representation theory, somehow complements Tom Hawkins' books. And there are other articles. For example, this article by Lam, representation of finite Groot 100 years, appeared in notices of the American mathematical society. His Conrad has on the web the origins of representation theory. And there is a book that, first coming, noticed that there is here a question mark. This is hopefully one day this thing may be published. But there is an interesting interchange of letters between Dedekin and Frobinus. Part of it appears in the other books that I mentioned before. But there is much more to be said about this. And there are some other articles more recently by Frédéric Breschenmacher about the controversy between Jordan and Kronecker and about the history of matrices. All of this is a very interesting and important historical material. Of course, I will not be speaking about all of this. I will just go into one important episode in this history. And then I will try to put it in a broader context of the history of algebra in the 19th century. And everywhere where people speak about the beginning of representation theory, there is this famous milestone, which appears in a letter sent by Dedekin to Frobinus on March 25 of 1896, in which he writes the following. On the whole, he says, one may well suppose that the properties of a group G regarding its subgroups will be reflected in the decomposition of its determinant theta. However, except for a trace, which indicates a connection between the number of ordinary linear factors of theta and the normal subgroups A of G, I have found nothing at all. And actually, it is quite likely that for the present, little will come out of the whole thing. So what is this determinant of the group that Dedekin is speaking about? Given G, a group of order n, these are the elements, G1 to GN, where G1 is the neutral element, we associate a variable x i to each of the elements g i. And we also call x stack i to the one associated with the inverse element. And we define a determinant theta as an homogeneous polynomial of degree n in n variables. And the variables are determined here in this determinant by the corresponding variables to the elements g i, i tag, et cetera. So you perform this determinant. You obtain that polynomial. And Dedekin tells to Frobenius that he thinks that there is a connection between the number of ordinary linear factors of theta and the normal subgroups A of G. And he says that if G is a billion, then that polynomial can be factored exactly into n linear forms over the complex numbers. So exactly in this way, here we have a product of these factors over here. And here we have an element not mentioned so far. It's not, you don't see it so clearly. Here it should be, I know it's OK. So here you don't see the hat, but it should be the factoring over all the elements of the group of characters of G, which was a concept that, as we will say a few words about it, was well known to Dedekin by having studied in several contexts. Now, Dedekin says in the letter that this result is a theorem which in this generality, as I believe, has not been announced as yet. But he also adds that if G is not an a billion group, then the determinant possesses, as far as I have checked, beside linear factors, also factors of higher degree that are irreducible in the ordinary sense. But this will be further decomposable into linear factor if one allows us coefficients beside the ordinary numbers, also hyper complex numbers. And he was referring here to the possibility of having factors which are non-commutative for the multiplication. And he added a non-proven conjecture if in the non-a billion case the number of linear factors going this way equals the index of the commutator subgroup of G. As I said, behind these ideas of Dedekin, there are many works, many contexts in which he was well acknowledged and had been working with similar ideas or stunning them. For example, Gauss' war on characters with finite a billion groups, which he used for assigning numerical properties to classes of binary quadratic forms. It had appeared in studying higher reciprocity and the Legendre symbol, also in Dirichlet application of analytical methods to number theory, and recent work on hyper complex systems that developed following Hamilton works on quaternions. And also, Dedekin's own work on number theory, I would say in the theory of algebraic number fields, which appeared, as you know, in the supplements to Dirichlet for Lesungen. In all of these places, we can find ideas that are related to this idea that Dedekin was suggesting here to Frobenius. And specifically, in the context of his work on the algebraic theory of numbers, or the theory of algebraic number fields, if we take k, a normal extension of the field of the rational numbers, and we take the Galois group of this field, then given n linearly independent elements of the field, he defined Dedekin in his work, the discriminant of the field as the determinant that you see over here, where you have these products of the linearly independent elements of the field multiplied by these elements of the Galois group. Now, from here, the idea was modified in the following way. If you take these elements to be the collection form when taking an element in the field and all its conjugates, given as above, then the discriminant that I defined before will turn into this determinant. Omega times pi 1, pi 1, pi 2, pi 1, et cetera. And from here, he went on to modify this idea into the idea of the group determinant, which is obtained when you simply ignore the element of the field. And then you get over here this determinant, where you only have the members of the elements of the group. And then with a little modification, you take only variables, x, y, j, instead of the numbers of the field or of the elements of the group. And therefore, you get a polynomial instead of a value. And with an additional little modification, instead of taking as the second coefficient, you take the inverse of the element corresponding. It's convenient because then you get on the diagonal the neutral element, which helps very much with the calculations and makes things easier. Dedekin also wrote to Frobenius in April 1896 in case you still want to deal with the group determinant. I allow myself to send you two examples that I thoroughly calculated on February 1886. And I quote this in order to indicate you that Dedekin have been thinking about these things for a long time. And this is very typical of Dedekin, indeed, because he usually was taking one idea and going on with it again and again and again. Sometimes he published, sometimes he didn't. But he was always rethinking and trying to produce more elegance. And as I will say now, we could say more structural ideas about the topics he was thinking about. Well, in 1896, Frobenius took the challenge posed to him by Dedekin. And this is the year where he publishes important works that are considered to be the beginning of representation theory, taking together or some specific results of the articles. And as you see here, we have the Fataushmare Matrizen. But we also have a work of group of characters and also about the prime factors of the determinant that Dedekin has suggested to him. So in these works, the first thing he did was to define the characters of the general finite groups. Specifically, he was defining the characters of the non-Abelian groups. The characters, I should say, these are functions from the group to the field of complex numbers that they preserve the multiplication. And in the case of the Abelian group, they are simpler because you can write the Abelian group as a direct product of cyclical groups. And then it's very easy to see that you have a finite number of characters defined in this way. In the Abelian case, it's more difficult. In the non-Abelian case, it's more difficult. And part of what Frobenius wanted to do was to come forward with this definition in the more general case. He proved the main theorems about them. And he applied these new concepts to solve the problem of factoring the determinant of a general group into irreducible factors. And before I go on, just a few words about the relationship between these ideas and the idea of our representation. If we take H to be the group algebra of G formed by these kind of elements over here, linear combinations of elements in the group with the coefficients in C, then we have a G-dimensional linear associative algebra over G. And there we can consider the following linear transformation, TG, which is simply taking the combination and multiplying each of the elements by the given element G. Now, if we represent this transformation using a matrix sigma G by taking a basis, for example, G1 minus 1, G2 minus 1, et cetera, then we can see without going into many details. But even the general idea that we have, that for every G or the mapping from G to sigma G gives us the right regular representation of G. And theta is the determinant written in this way, where each element or each variable is multiplied by the sigma for G1, G2, up to Gn. What happened here? And moreover, sorry, something moved with the slide, but you can see here, moreover, that if M is a non-singular times N matrix over C, such that it can be that the product, this product over here, M, sigma, M minus 1, yields a matrix of this form with two blocks, one R on R and the other S on S, then the determinant here results as a product of two other determinants, each of which is of polynomial R and S. And hence, we can see also that the composition of regular representations into irreducible representations is equivalent to the decomposition of the group determinant into irreducible factors with corresponding degrees. So the relation is quite straightforward. And Frobenius formulated the problem in the following way. If we have a factoring of theta into factors in which, in this case, we have here factors going from 1 to L, each factor we call in phi lambda appears with a multiplicity, or as many times as E lambda. If this is the factorization of the determinant into the different irreducible factors of each of which is of degree F lambda, then as Frobenius, how does this factorization represent the property of the group? And remember, this was the question posed by Dedekind. And as I said, with the kind of new tools and new ideas that Frobenius introduced in 1896, he formulated the problem in this way, and he also came up with solutions and new problems, new ideas. But basically, he was also able to address two specific questions he had in mind. First of all, whether the number of linear factors here is the one that Dedekind had conjectured. And secondly, whether the various parameters that I obtained here, L, E lambda, F lambda, whether there is a clear relationship between them, NG. So as I said, he investigated in those three articles the various new ideas he had, symmetric matrices, and applied them to the properties of sigma G, which, as I said, is the matrix representation of that linear transformation. And among other things, he came up with the following answers. First of all, the Dedekind conjecture was correct. Secondly, L equals the number of conjugate classes of G. Third, E lambda equals F lambda. And that means that each factor in that factorization occurs as many times as its degree. And as I said, he also generalized the concept of characters and some of the basic results about representation of groups. OK, now the background of Frobenius to this kind of work was quite different to that of Dedekind, or had different elements from that of Dedekind. Because Frobenius in the previous year had been working on topics like the theory of linear differential operators, linear forms with integer coefficient. He had provided an improved proof of Silo's theorems. And he worked, especially important for this topic over here, on linear and bilinear forms, and on the theory of the quadratic forms. And this is certainly one of the reasons or one of the circumstances against which we have to see why or in which way he took the challenge and developed it. But there is more, I think, to the difference between the backgrounds of these two mathematicians, which tells us a lot about the way or the main threads in the development of algebra in the 19th century. I just put, again, the topics that I mentioned in relation with Dedekind to remind you the different backgrounds. But we may ask in a more specific way over here whether it would be natural to expect that someone like Frobenius would like to produce such an idea of representing groups via matrices and, hence, trying via them to have new information, additional information on groups. And I say, well, seeing it retrospectively, we say, well, it's a natural thing to do. It's something very efficient, very fruitful. Why not? But one point that I want to stress here is that if we look at the opinions of several people working on group theory, on groups, I would say at that time would say that it was not very clear what were the kind of questions to be solved in relation with groups, and specifically not that one way of representing them or of trying to find out the properties would be the convenient one to do it. Let me give you some examples. But to put it in a more specific way, actually, what I am asking is what was group theory in 1896, what was algebra in general in 1896, and even what was group theory for Frobenius or for Dedekin, and what was algebra for Dedekin or for Frobenius, and even something to be related with the topic of this colloquium, how is all of this related with the work of Galois? And the point is that at this time, it is far from obvious, and it is far from uniform, the way in which these questions were answered. For example, if we consider Cayley speaking about groups as group of permutations, he says in 1878, the general problem of finding all the groups of an order n is really identical with the apparently less general problem of finding all the groups of the same order n that can be formed with the substitution upon n letters, permutation groups. This, however, in any wise, shows that the best or the easiest way of treating the general problem is thus to regard it as a problem of substitutions. This is not the way. And it seems clear that the better course is to consider the general problem in itself and to deduce from it the theory of group of substitutions. Let's just say we are able to represent the group in a certain way, but it doesn't mean that if we want to find out the properties of groups, we should look at it in that particular way. No, I know to do that, but I will go the other way around. Weber, in his famous book on algebra, 1896, speaks about groups as groups of linear substitutions. The importance in algebra of linear substitutions, and in particular of the finite group that they define, concerns the fact that groups of permutations of n elements can be represented as groups of linear representations. And then in another book, Burnside in 1897, in the theory of group of finite order, Cayley's dictum that a group is defined by means of the laws of combination of its symbols. That is the fact that Cayley is saying that a group is we must not consider the nature of the element. Just look at the laws. Would imply that in dealing purely with the theory of group, no more concrete mode of representation should be used than is absolutely necessary. It may then be asked why in a book which professes to leave all applications aside, a considerable space is devoted to substitution groups, while other particular modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with the properties of substitution groups, it would be difficult to find a result that could be most directly obtained by the consideration of groups of linear transformation. So you see people saying, we should do it this way. We should not do that. It's not clear at all. But OK, we're in a time when many new results are being found in group theory, and in other parts of what we now call together under the title of algebra. So going back to these questions, we see that the answer, historically speaking, is not straightforward. But van der Daden, in his famous book on history of algebra, from Al-Juari's Mi Tuemi Neter, he puts things more straightforward than this. And he says, modern algebra begins with the Varis Galois. With Galois, the character of algebra changed radically. Before Galois, the efforts of algebras were mainly directed towards the solution of algebraic equations. After Galois, the effect of leading algebras were mainly directed toward the structure of ring fields, algebra, and the like. And I think that if we look at the developments, including those that I have just mentioned, on Dedekind and Frobenius, one has to take this formulation in a very different way. It's not as straightforward as it appears here. And it's curious that one who plays an important role as a mathematician, not a certain historian, is van der Verden himself, of course, with his famous book on modern algebra. And I think that by looking particularly at this famous map or light fountain appearing at the beginning of the book, by looking at this, we can understand better the kind of development that I am talking about that was not as straightforward as van der Verden the historian says. In the sense that what we see here appears here, I would say, for the first time in this clear and crisp form, in the sense that here we have all the basic structures of algebra, groups, polynomials, fields, rings, et cetera, they appear here for the first time in this very schematic and, I would say, equalitarian way, saying that all of these structures are individual instances of one general idea that is the idea of an algebraic structure. And simple as it may sound, what we see in the development of the ideas related to algebra is that this idea was not found so early as following Galois. And it was not found in 1896, so clear as it is here. And many developments had to occur before we reached this clear situation. Here, notice that the idea is not just that they are instances of a same idea. By being so, we have to investigate them with similar tools, with similar approaches, with similar questions, with similar answers. And all of this, I think that even from the few examples that I gave you about group theory in particular, not to speak about the other topics, it's not something that is found yet. And it has yet to happen when we are in 1896. So between Galois, between Galois, what happened? Between Galois and van der Verden, between the situation in which people start to speak, Galois start to speak about groups. And the situation in which we see algebra as being devoted to studying structures, there are many things that happen in the middle. And I would rephrase it this way. This is me quoting myself. With Galois, a long and complex process started, which eventually led to a radical change in the character of algebra. The efforts of the leading algebra and the algebraists were increasingly directed towards the structure of Ringfield's algebra and the like, but there were other things going around. And some of them also remained after van der Verden. Finally, with Netter and with van der Verden books, which helped promote the ideas of Netter, the idea that algebra deals with structure was consolidated, and it became very influential on subsequent developments. Among the things that happened before that could crystallize in that way, of course, they're important milestones, like Einstein's article mentioned yesterday. And it is important because for the first time, it proposed that fields should be investigated in the abstract and using the same or asking similar questions or questions that are similar to those that we use when investigating group theories. You will see, I will mention a couple of examples to show you that this was not so clear at the time. Interestingly, the main trigger for Steinitz before formulating this came from the theory of piadic numbers. Piadic fields where we had fields with characteristic d. And of course, it had very little to do with the developments within algebra. It came from a different direction. So we have here the abstract theory of fields. On the other hand, we have the abstract theory of rings that Netter helped formulate partly against the example of Steinitz, and partly building of things that had come earlier to her. And as Colin mentioned yesterday, she used to say a state alles shown by Dedekin. But I put it here with a question mark because I think that nicht alles, a lot of it, but not everything, and not everything, I'm not talking so much about the technical details, which of course people, including Netter herself, had to still to work out. But also in that basic idea that I think that can be graphically be seen in van der Werden's light fadden, it is interesting to see that even in a mathematician like Dedekin, this is not found in this way, and certainly not in a mathematician like Frobenius. What I mean is, for example, that if we look at Dedekin, for example, in his lectures on Galois theory, mentioned in one of the earlier talks, yes, he took the ideas of Galois and started to develop them in a way that comes much closer to what we know nowadays. However, look at the following interesting difference that at least I see while reading them. On the one hand, he has the groups of substitutions. On the other hand, he has the rational domains. So, seen from our perspective, here you have one kind of algebraic structure, here you have a different kind of algebraic structure, and now you start to see the relationship between them. But this is not really what happens in Dedekin, in his lectures, we know them from the manuscripts. Rational domains is a subset of complex numbers which is closed under the four arithmetic operations. So it's a specific subset of the complex numbers. Groups of substitutions, we studied them because the properties of the groups, no, sorry, something happened here, we have the rational domains, which are a subset of complex numbers. And the group of substitutions, we in this case, Dedekin, before even he starts his entire discussion, he gives a very systematic and complete, from his point of view, overview of the properties of the groups. And this is not the case with the rational domains. Here he starts to work on them, and the properties are briefly discussed whenever needed. OK, you may say, well, this is just a difference in the way he's exposing the idea. I think there is much more about this. And we can see it in the following way. The system of numbers that he's dealing with in his lectures on Galois theory, the systems, this is the subject matter of higher algebra. And this is the thing that we want to study as part of the theory. And the groups are a kind of innovative, efficient, we know that's rather abstract tools that we use for discussing the subject matter, which is the domains. So what I mean, they are not put on the same footing. They are not treated equally. And they have a different role and a different essence from the point of view of Dedekin. Another example where we can see this is in the theory of algebraic number fields that you may know Dedekin published in a series of versions. There were four published versions. We know that, as I said there before, regarding the other ideas, Dedekin was always thinking about these things and changing them. And the changes have to do with the fact that for him the basic concept that he's using, the theorem and the proofs, we're talking mainly about proofs of a unique factorization, theorems are successively formulated in a way that every time he's able to avoid more and more, not totally but increasingly, the need to choose specific elements, for example, for defining what is a prime ideal or for proving a certain uniqueness theorem and so on. On the other hand, he has the ideals and the modules. And again, the ideals and the modules in this work are not the subject matter. They are the tools. They are the tools with which you are going to investigate the properties of a unique factorization and so on. So it's similar to the difference I made before between the domains and the groups. Here you have the fields and the ideals. Again, for us, seeing it retrospectively from the point of view of Van der Verden, these are various algebraic structures. So again, here we have the fields and the algebraic system and their interrelation constitute the subject matter of fire mathematics and the ideals and modules are the tools. And it is interesting to see how Frobenius considered, how he evaluated the work of Dedekin in a letter to Weber in 1983, when Weber was about to write his book. He says, he tells him, your announcement of a work on algebra makes me very happy. Hopefully you will follow Dedekin's way, yet avoid the highly abstract approach that he so eagerly pursues now. His newest edition of the Fort Lesungen contains so many beautiful ideas but his permutations are too flimsy and it is indeed unnecessary to push the abstraction so far. I am therefore satisfied that you write the algebra and not our venerable friend and master who had also once considered the plan. I think that this is very indicative of the kind of different views that are going on around here. Of course, we are speaking here of Frobenius, a representative of the Berlin School with Kronecker and Weierstrass, but mostly under the influence of Kronecker who are their approaches more, as Harold Edwards explained about Galois, but he has written a lot about this on Kronecker, more of calculation of choosing specific, of basing the concept of a specific choice of an element and there is a dialogue between these two kind of ideas and it comes very nicely also in this encounter between Dedekin and Frobenius on the creation of a representation theory because Dedekin has this idea but it doesn't go so much in the direction of the kind of things that he's doing and he suggests it to Frobenius and there it catches very nicely and very naturally with a point of view that he had been developing for what we call today algebra in general. Notice, however, that in the background that I mentioned about Frobenius, there is no specific, before there is no specific article about matrices and this is an interesting topic in itself because all the idea of matrices, again, if we look at it from the point of view of the scheme presented in van der Werden's book, well, it's easy, but we have matrices, it can be in linear algebra, it can be an example of a ring, et cetera, but at this time, matrices are in a phase of transition. Usually ideas are always in transition but here we have a very clear moment of transition because on the one hand, you have the roots of the idea, for example, in Cayley and some other British authors where it appears more as an extension of the idea of an hyper complex number. You saw that Dedekin, at least in the text that I cited, he doesn't speak of a matrix, he speak of a determinant. And you have Frobenius himself having worked with the bilinear forms, et cetera, having developed the idea from there. So the idea of matrices, it doesn't have a clear place in all the map of the algebraic or what we see as the algebraic ideas in retrospect and is really in a constant movement over there. And in this movement, well, we can see, for example, that Emineter didn't like it very much because she's developing the way of Dedekin, let's say, as expressed. And here we have a very interesting testimony by Dubray who here I will read in French. I hope not to be embarrassing. Le course d'Emineter n'est pas facile à suivre. J'ai un jour une difficulté à propos d'une affirmation qui ne m'est pas restée pas justifiée ni dans son course ni dans son mémoire. Une démonstration s'est obtenue sans peine pour une calcul de matrices, mais j'étais devenu assez nétérien pour ne pas m'insatisfaire. À la fin de la question suivante, je suis allé poster la question d'Emineter qui répoussa énergiquement les matrices et après trois secondes de réflexion, mais montre combien la show était claire si l'on jonglait à doigts mains avec les modules. I think this is very typical and it summarizes in a nice way the developments I have trying to indicate precisely because Netter certainly was developing so many ideas of Dedekin, but was bringing them to this new position in which they will appear in a way that all these ideas, groups, modules, ideas, fields, rings, they are all citizens with equal rights in this republic of algebraic structures and within this she would rather not consider arguments coming from matrices where you multiply with specific elements, of course she would prefer the chain arguments, et cetera that had appeared already in the work of Dedekin, but here they came to friction in the way that she defined it. So that in the end I think that the conversation or the interchange between Dedekin and Frobenius, it's important not just because the specific contribution it made, that it triggered, that it prompt Frobenius to develop the ideas, maybe he has reached them otherwise, but the fact is that this is the way it happened. So it's not important only because of that, but it's important as a way to look back at the history of algebra in the 19th century and to realize that there were many complex, sometimes conflicting and sometimes cooperating views and in a way here you have it in a nutshell that helped us understanding it very clearly. Thank you very much. I would like to ask you and see quickly there was an arrow, there was an arrow going from Galois to modern structures at some point, yes? Do you consider this arrow if you may allow me to use the expression as a walk to the holy land or would you rather say that now for example if you look at the following period then we are more in a backwards arrow or different direction? Well, this is history, not mathematics, so things are not necessary. Things may happen one way, may happen the other. And there are certain circumstances that help us understand why this idea caught so strongly. Well, the first reason is that mathematically speaking it was a very good idea. I mean mathematics and algebra specifically became very fruitful, people started to work and to produce many ideas and so on by looking at algebra in this way. So because if you take after Van der Verden then you have other books like Birkhoff-McLean in the United States and there are several other less influential books but again they brought this idea that this is what we have to do in algebra. And of course Burbaki later on taking this image and extending it to the whole of mathematics. And many people entered this and worked and produced important mathematics. At the same time however, there were people, I mean not everybody was working algebra in the way Van der Verden did even though well it got very strongly and most people were doing it. And I think that like in many other episodes in the history of mathematics ideas at some point became less interesting to new generations and they start to try to look at different ways for example, okay nowadays we have very different trends in algebra. Not the person to speak about them but it's obvious even if we speak about computational trends and so on so that I think one of the interesting things historically looking at this that is clear for many people working in algebra at the time and then with Burbaki is that came an idea that was perhaps new in mathematics that here we reach the end of the story. So this is how things will remain forever. You will have more results, you will have more of the same, you go on. But this basic idea what is algebra and what you have to do in order to pursue algebraic research, this is here to stay. And I think it's nice that this is not the case. We are moving all the time and I guess that you know it's like a pendulum. It will come back in some way, but the specific situation is one that was once in history and will not return exactly like that. Okay, yes, yeah, okay. And you're quoting people taking it seriously. Yes. In the circle that I was brought up in, it was clear that this was a catchphrase. One of her favorite catchphrases came out often. It was a coffee table phrase. It was a what's nowadays called a soundbite in politics. And it had just as much and just as little meaning as any political soundbite. Would you like to comment on that? Yeah, sure. No, I think it has more meaning than that. To a large extent, I agree with Colin who took it very seriously yesterday. You know, if you look at Dedekind, his work on algebraic number fields and you see how, this is an interesting thing that you have the evidence, that you have various versions and you see how he works out the ideas so that you know, for example, that the most important one in this context is the chain conditions. They don't appear in the early versions, but they appear very clearly in the later ones. So you don't have perhaps all, you know, for example, you don't have the concept of a ring in Dedekind, in the sense that here is a ring, here is an ideal, and if you have a chain of ideals of that kind and that kind, you will get this and that. But the fact that you are going to analyze the unique factorization theorems by looking at the way in which ideals are contained in each other and the properties of this thing, this is quite clear in Dedekind. Not perfect, not with all the details, but it's there. So Alice by Dedekind is a niche Alice, but many, many important things are there and I think she meant it quite seriously. That it convinced me to go and look for it. I say, no, well, I'm supposed to be an historian but I'm supposed to be careful about these things and not to find in the past things that are all in the present, but I think I found some of them there. Yeah. I want to speak on that. Netter had all her students read all the appendices by Dedekind in every version from the first to the last, all the appendices in every version. She had her students read it, it's not a sound bite. You disagree, but anyway. That's what we are here for. Yes. Yes. A quotation from the first 1896 letter. Yes. The letter of 25th of March, I think it was. Okay. 1896. That's the one. Yes, a connection between the number of ordinary linear factors of theta and the number of normal subgroups A of G. No, and the normal subgroups, not the number. I'll tell you, it doesn't say the number of normal subgroups. Sorry, I beg your pardon, it doesn't say the number. No, the number of ordinary linear factors of theta and the normal subgroups A of G. Yeah, well, I think this is. I mean, if we would phrase it today, we would say between the number of ordinary linear factors and the structure of G. So what he is doing, what do you want to say about this group G? Basically, this is being structural about G, saying the relationship between the subgroups, et cetera. So this is his way to saying that, I don't think that he's meaning some specific property of the normal groups. Whatever you want to say about the normal groups, how many of them you have, et cetera, or what are the connections? I think this is the idea here. Right, so this is not a mistranslation or anything. I don't think so, I hope so. Okay. Yeah, no, it's not a mistranslation, I think. Yes, I have a question about group morphisms. Because we now tend to look at representations and characters as examples of group morphisms. But it's not entirely clear to me if they did at the time. For instance, their issue, as you showed, was one of representation. Do we want to represent the elements of a group in some other way than just letters? Do we want to have matrices to represent the elements of the groups and work with these matrices? So is there a problem, just one of more concrete representation, or do they construe things as having something to do with group morphism? Because group morphisms were seen in a quite different way, in terms of usually generators and relations, and you add relations and you have an isomorphism, a mergédrique. So the idea that when you have a group, you also have group morphisms that come along, which is now so commonplace for us, was not commonplace at the time. No, yes, I mean, you're right about morphism, and this is one of those ideas that finally, in Van de Verden's book, become very clear that you are looking for isomorphisms everywhere, or homomorphisms with a kernel or whatever. This is the kind of thing that I say, you move from one place to another, even though it started in a specific context. And I don't remember exactly where these things, I mean, the idea, it's in some of the quotations here by Cayley and so on, you now have the idea that things that may look different are basically one and the same group. So this idea of isomorphism, but it's not so clearly defined as would be later, precisely because you have one in each. I think that in the case of representations, this is not the case. The case is more than trying to translate into something which is not the same thing. Well, it's strongly connected, whatever, but in a place where it's easier to find some properties, et cetera, which, well, it's an old strategy in mathematics, like you do it with geometry and algebras in the cart, and you try to do it in other places. Now, because both things, one is a group and the other thing is a matrix, and we now see it as part of the same thing that you could tend to see it more as an isomorphism, but basically, no, it's moving from one domain to another. Both of them, by the way, not very clear, because it's not yet very clear what it's theory of group and it's not yet very clear what is a matrix or linear transformations or whatever, but you have the idea or they have the idea that perhaps here it will be easier to find some of the properties. I would like to first point out the truism, which will be one to you, but still it's worth saying, and then ask a question. The truism is that when you look at the Deakin, Connecker, and to some extent Weber, although I haven't read the whole algebra, they don't pursue the algebraic properties to the end, but the reason is simple, they didn't care. Their purpose was the queen of mathematics, namely number theory, and they developed the means, they know what a module is, they know what an ideal is, of course, but that's not an object for itself. They just do only what is needed to prove the higher number theory. Sorry, and the change comes when the tools become the object of study. It was always, but it's not one-zero situation, but it's a focus, the focus moves to the tools. So that's the truism, but it's very important when you read them. The second thing is, despite the adventurer and the ife head, there is a Frobenius substitution. Okay. So can you say a few words about this? No, I don't know. You know, I mean, I have no idea. I found under Frobenius substitution, the one with the heads. Leo, from the genealogy you described, there was first the group determinant, then it's linear factors from which came the characters and only later the representations and the connection with matrices. In this, yes. Do you think that the fact that Frobenius found matrices useful for this purpose was one of the things that sort of authenticated them to the mathematical community? Because it seems that before this, there were some curious objects. They is the matrices? The matrices. It's, I guess that it certainly helped, but the history of matrices, I mean, I wasn't very aware of this. I must say I read more and more in the articles by Frederick Brechenmacher. And also it appears in a way also in Tom Hawkins, but in Tom Hawkins is so densely packed with the league groups, et cetera, that it's hard to see. And what I want to say is that this is, if you compare the history of matrices with the history of groups or ideals or whatever, you have something much more confusing and because the roots come from all around the place, from analysis, differential equations, bilinear forms, hyper complex numbers, and so on. So that it may, even though I would think again, because one should see how long did it take for representations to become a more canonical object, a more canonical discipline, let's say, right? Because it's not so easy. I mean, Frobenius comes up with the articles I mentioned here, you have to wait a little bit, then the other people, like Burnside and Brower and so on. And there are some gaps in between. So it helps, but I don't know how quickly and how strongly. I don't know the story so closely that I cannot tell you. So my question, well, one question is, is there any qualitative difference between Frobenius's kind of complaint to Weber and any of the other complaints that we, always exist about some mathematics being too abstract. I mean, now it's easy to find complaints about higher category theory, complaints about growth index, complaints about how to talk about Boba Key. So always seems to be complaints about algebra. About the abstract, people going to abstract. Yeah, I think you can find it even earlier than that. I mean, two things. One is that the abstract of today is the concrete of tomorrow. And here we have an excellent example of that because groups becomes the concrete in category theory or any of the other. And it simply, I think it shows that at any given point in time, people are thinking differently. And we sometimes, when we specifically, when we think something that got so strongly as van de Verden, we may tend to dismiss all of that as people who, they were wrong, simply. But they were not simply wrong. They had a different idea, a different take on that. And for certain reasons, in some cases, we can explain historically, one of the views became stronger than the other. But I think you yourself mentioned it. You have these kind of tensions all around there. In that sense, this is not specific to Frobenius and Dedekin. What I think it's interesting that Dedekin at the time, he was quite isolated in his views. I mean, people didn't like very much to read Dedekin. Even his friends, even people who thought, even Frobenius, who thought our venerated man, I think he means it. But will he start to read all these nuance things about the idea? Probably not always. So the interesting thing is how this thing that was a tiny minority became the majority view. It has to do a lot with Hilbert, with Netter, and things like that, the getting in trend, and so on. I would like to ask you about a different direction that abstract algebra developed, because you've been talking about the growth of abstract notions in algebra based on very concrete issues of solving equations and doing permutations and so forth. But there is also, in the 19th century, a direct connection between abstract ideas and modern algebra, which is, of course, George Bull's laws of thought and modern logic. The structures of modern logic as developed by Bull and other people after that is not based on anything sort of concrete in terms of solving equations, but based on, as he himself said, the laws of thought. Could you comment on the way that this direction fed into the development of modern algebra? Yeah, well, this is we can perhaps subsummit under the title of algebra of logic, which starts with Bull, then you have Grassman, and many other people, Schreder. It influenced it, but I think that it influenced it less than one might think. In the following sense, if we try to understand modern algebra only by looking at the fact that some of the concepts were defined abstractly, like Cayley with groups and so on, then if that was all what was the development of modern algebra, then all what happened in that direction would be highly important. Because then you say, well, basically, mathematics is just about abstractly defined things. But what I am trying to show, among other things here, is that this is not enough. You need something else. And the something else is the identification of several ideas that were formerly seen as separate to see them as being actually one thing. One thing that is not defined, it's not the case like in mathematics. You define group formally, and then you see it's the same, this and this and this is the same. This is not what I'm talking about. I'm not talking about the formal concept, which, like in Burbaki, appeared structure. This is not the case. I'm talking about a general idea that says all of this is the same. And actually, what happened with the ideas you mentioned, they were added later on, like saying, well, you have here Boolean algebras. I don't know. This is another instance of an algebraic structure. So logic is nothing but algebra. But it came relatively later. You don't find it in Van de Verden. You find it in other books that came out later on. I have a few remarks which sort of support your thesis. Yes. The first is you mentioned Lam and his history. But he also has a nice little note on Frobenius. And he characterizes him as a very forceful character. And I'm sure he was. Yes. And you also mentioned the sort of slight difference between gutting and burlin. I think that played an important role. It does not only go back to Kronika, but who was the successor to Frobenius? He was sure. And I don't think Schuer built great theories, but he did prove theorems. And somehow, this was a serious controversy between these two universities. Also financial support from the government was entailed, right? But it was also a game. It's a bit like the boat race between Oxford and Cambridge. I mean, they are fierce competitors, but it's also fun. I completely agree. Although one must say that there were periods. And in this sense, Frobenius is taken not to represent the golden age of Berlin in the sense that were a few students, et cetera. And he seems to have been a difficult person. Actually, when reading his biography, I learned a new word in English that I didn't know, Krantakeros. Because he's described as Krantakeros in more than one place. So now that I read that biography, I know what the word is. And probably he was not a nice person. But the others were not so nice either, one must say, except Nederman. I would like to ask a question. Where is it? Whose answer is centrally well known to the experts, but I'm just a humble working mathematician. So in your lecture, you mainly concentrated on the German school of algebra. But weren't these results in the representation theory also many of these independently discovered by Burnside? Yeah, I thought you were going to ask about France. No, I'm not French. Yeah, I mean, I specifically said the story is, fortunately, I was asked to speak about Dedekin and Frobenius so that I can focus on that. But there are, I cannot tell you exactly if the word simultaneous is very precise, not in terms of time and not in terms of independence. You know, I can refer you to one of the books by Curtis or by, it happens at the same time. And then you may ask, well, what happened there? He didn't have his Dedekin to suggest. Well, of course, I mean, there is no big surprise in any of the things that I presented here. It's working out ideas. I think the big surprise is the ability to put together things that came from different sites of the map. And perhaps if I looked closely at Burnside, which I didn't, I could tell you what threats he was putting together over there. Yes? They are saying no. OK. No, Burnside always. Oh, OK. Yeah, by the way, this is another important thing that all of these things are proved, the use of them appears when they are used for some useful proof. Otherwise, and with structural algebra, it happened because it helped solve many problems to formulate more clearly uniqueness to your aims. And so otherwise, it wouldn't have taken the way to. I have my door, I think, to the question. Thank you. Thank you.