 Welcome back to our lecture series, Math 42-30 Abstract Algebra II for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In this lecture number three, I want to talk about the very important topic of automorphisms. What is an automorphism? Well, it's appropriate we start with the definition. Well, essentially, an automorphism is just the marriage of two topics that we've explored previously. Well, at least we explored it in Math 42-20 Abstract Algebra I. Anomorphism is an isomorphism, which is also a permutation. Remember the word isomorphism itself. We have a lot of words like isomorphism, homomorphism, endomorphism, epimorphism, monomorphism, right? Morphier means shape, right? So, homomorphism, a morphism of some kind is a function between two algebras that's measuring the shape of the things. Isomorphism here would mean they have equal shapes. Two groups are isomorphic if essentially they're the same group up to relabeling, of course. Anything that's not preserved by isomorphism is something that's basically irrelevant with respect to group theory. A permutation as we've studied many times before in Math 42-20 Abstract Algebra I, a permutation is a bijection from a setback to itself. It scrambles everything up potentially. An automorphism is the marriage of the two things. More specifically, an automorphism is a map from a group G back into itself, which is both bijective and homomorphic. So, an automorphism is an isomorphism from a group back into itself. That's actually where the name comes from. We see this term morphism like we've seen many times before. So, this is a map between groups that preserves the multiplication. But here, self is the word auto. That is auto means self. Like an automobile is an object that moves itself, it's self-mobile. An automorphism is an isomorphism from a group back into itself. Of course, you have analogous ideas for like automorphisms of rings and such. For any algebraic category, we can talk about an automorphism. An automorphism of a group is essentially the symmetries of that group. It's important to study these. Automorphisms are a very important topic in abstract algebra, particularly in group theory. I kind of let the Canada back there when I talked about symmetries. If we take the collection of all automorphisms of a group G, this is commonly referred to as Aught of G, A-U-T of G. And this is called the automorphism group of G. Although the name seems to suggest that the collection of automorphisms itself is an automorphism. Excuse me, the set of automorphisms is itself a group, which it is. And as I mentioned with the symmetry statement, we really wanna think of the automorphism group of a group. We want to think of that as the symmetry group of the group. And I can get a little bit confusing, I confess, because we're talking about groups that are derived from other groups, but this is common practice in algebra, of course. So let's actually give a proof of this statement. It's very straightforward. This, that is, if you give any group whatsoever, an abelian group, non-abelian group, finite group, infinite group, doesn't matter. Any group whatsoever, its collection of automorphisms will form a group. In particular, we can view this automorphism group as a subgroup of the symmetry group, S sub G, where this is the collection of all permutations on the elements of G. These are not necessarily permutations that preserve the algebraic structure of G. It's just permutations on G as a set. And that's why odd G is so important. This will then be the largest permutation group that preserves the algebra of the group, the algebraic structure there. And so this is actually a very straightforward argument, but let's spill out the details for us here. So as we've seen before, if we take the composition of two isomorphisms, this is itself an isomorphism. So if I have two isomorphisms, say phi from G to G, and we have say psi from G to G, then it's very clear that phi composed with psi will be an isomorphism from G to G. So if phi and psi are both isomorphisms, then the composition will be isomorphism. The composition of two bijective functions is a bijection. The composition of two homomorphisms is a homomorphism. So hence the composition of two isomorphisms is in fact an isomorphism. Well, for an automorphism, you need the domain and co-domain to be the same thing. And so when you compose these together, you're gonna take the domain of psi and the co-domain of phi. So the composition of automorphisms is likewise gonna be an anomorphism, okay? That's really what we need to get out there. Because again, the composition, like I said, isomorphisms and isomorphism, the identity map between two groups is an isomorphism, the inverse of an isomorphism and isomorphism, these are all facts we've established in algebra one. Similar statements also are true for permutations. That's why permutations make a group. The composition of two permutations is a permutation. That is the composition two bijections is a bijection. And of course the domains and co-domains are what they're supposed to be. The identity function is a permutation. The inverse of a permutation is a permutation. So when you marry the two ideas together, isomorphism and permutation, then the compositions have all the things we need. It's very trivial to see that the collection of automorphisms forms a subgroup of SG, okay? And so that really is the proof. There's not much more to say than that. We've already done all the pieces. We just have to connect it together with the vocabulary in hand. And so before ending this video, I wanna point out that the automorphism group acts on the group G itself with a very natural group action. If you take any automorphism in G and any element of G, then how does the automorphism sigma act upon G? This gives you sigma of G. So it's just the permutation action, right? We've learned previously that if a permutation group acts on a set, you can just do function evaluation to be that action. And so given a group, it's automorphism group, which is itself a group, acts upon the original group. And this action is actually a very important action. And in fact, we oftentimes when we study group actions when you're acting on a group, we ask, is it acting via automorphisms? That is the group action really just a subgroup of the automorphism group right here. And these are important, right? The construction of a semi-direct product depends upon this action right here. We can only take a semi-direct product between two groups if the one group has a homomorphic image inside of the automorphism group. Some of these details we have not talked about, but I'm just trying to allude towards the importance of this automorphism group. And in the subsequent videos for this lecture, we'll do some examples of automorphisms of groups.