 So we've created a bunch of algebraic structures and found a lot of properties, so let's try and summarize what we've found. Now if you have an amorphous blob like a magma, there's not a lot we can do, so let's talk about monoids. A monoid with the Latin square property has closure, identity, associativity, and if finite, inverses. Now the Latin square property is a useful way to analyze algebraic structures as long as there aren't too many elements. But it's tedious to verify, and so the question we've got to ask ourselves is, can we use a different property? So let's think about that. Suppose g star is a monoid where every element a has an inverse a inverse. Now suppose a star x is equal to a star y, then since a has an inverse, we can left multiply by a inverse. Since g stars are monoid, we have associativity. Since a inverse is the inverse, then a inverse star a is the identity, and since it's the identity e star x is x, and e star y is y. And what this means is that in the Cayley table, no two products in the row of a can be the same. Since in order for the products to be the same, the second factor must also be the same. And by a similar argument, no two products in the column of a can be the same. And so this means that if g star is a monoid where every element has an inverse, then g star satisfies the Latin square property. And there's one important thing to notice here. Previously we've had to require that g have a finite number of elements, but notice that we don't need to assume g has a finite number of elements anymore. And so this leads to the following observation. Having the Latin square property and a finite number of elements gives us inverses. But having inverses gives us the Latin square property whether or not we have a finite number of elements. And since the inverse is easier to check, we'll use it as the defining property of a new algebraic structure. And so we define a group. Let g be a set and star a binary operation where g is closed under star. So that gives us a magma. Star is associative. That gives us a semi-group. There is an identity element e where for any a and g, a star e is e star a is a itself, which gives us a monoid. And finally this last property. And every a and g has an inverse a inverse satisfying a star a inverse or a inverse star a gives us the identity. Then we say that g star is a group. Now really the group idea is a culmination of the different things that we've developed. And we've already proven, but you should actually try to prove these from the group definition. So in the following results. The identity element of a group is unique. The inverse of an element in a group is unique. And groups satisfy the Latin square property. And let's put everything together. Remember we want to be able to solve equations. A group is the simplest structure where the equation a star x equals b always has a solution. And the solution is unique. So in some sense a group is the simplest structure we can do algebra with. So let's do some algebra.