Basic Algebra – Some Group Theory Theorems

So, now that we’ve seen some of the basic notation, I’m going to introduce some of my favourite group theory theorems.

I’m going to start with the obvious: The First Isomorphism Theorem. This is a classic for a reason, namely that it can be used to prove pretty much every other result ever (maybe not quite, but almost) in group theory. It’s also a very simple, beautiful result. The statement is as follows: Let G and H be groups and \varphi: G \to H a homomorphism. Then Im(\varphi) \cong G/Ker(\varphi).

This theorem is useful whenever a you have to show that two groups are isomorphic. Simply construct a map such that one group is isomorphic to the image of the map, and the other group is isomorphic to G/Ker(\varphi). The easiest way to deal with the right hand side is to construct \varphi such that the kernel is trivial. As I’m sure you can see, this is a very useful theorem, and the other two isomorphism theorems follow from this one, and so does the correspondence theorem.

The next theorem I’m fond of is Lagrange’s Theorem. It’s a very pretty result, that can be shown easily by reasoning with cosets. It states that for a finite group G, the order of any subgroup H divides G. The proof actually shows slightly more: namely that the quotient of the order of G and the order of H is the index of H in G.

I like this theorem because it has some applications in number theory. It can be used to prove the Euler-Fermat theorem, and a neat result called Wilson’s theorem. Note that it states that any subgroup has an order dividing the order if the group, but it does not say anything about there existing a subgroup of any given order. This next theorem does though:

This is, as with the Isomorphism theorems, a collection of theorems called the Sylow theorems. They are named after a Norwegian mathematician called Ludwig Sylow and tell us about the order of some subgroups of a finite group. They assert a partial converse to Lagrange’s theorem: that for any prime p and natural number n such that p^n divides the order of the group G, G has a subgroup of order p^n.

The subgroups of prime order are called Sylow p-subgroups, and the second Sylow theorem states that the p-subgroups are conjugates and hence isomorphic. The theorems can for example be used to show that a group is not simple, by classifying all the subgroups.

These are all very basic theorems, that one should come across during an undergraduate course in group theory, and together I think they give a fairly good idea of the basics of group theory.