Point of Post: In this post we discuss the theory of -groups acting on sets, and some of its ramifications.
I have previously discussed group actions, but being a rush to discuss group theory I skirted over some of the beautiful theory. So, I’d like to take some time to discuss one of the prettier and more powerful branches of the theory, namely when we restrict our attention to group actions by -groups. Not only will we be able to say some prove some fairly substantive theorems about -group actions explicitly, but will be able to prove some very neat things in more general group theory and in number theory. The interesting fact about the theory we will discuss is that at the root of everything is a ‘fundamental theorem’ whose presence (being the theorem in the case of a particular group action) is the proof that every -group has a non-trivial center. Namely, we were able to conclude that since the cardinality of any conjugacy class must divide the order of the group, that they must be divisible by . From this and the fact that the sum of the cardinalities of all the distinct conjugacy classes must sum to the order of the group (which is divisible by ) that the sum of all the one point conjugacy classes must have cardinality divisible by . Well, the generalization of this idea (which, as I’m sure is pretty clear, can be restated for an arbitrary action with conjugacy class replaced by orbit) will be the main tool I mentioned from which all our other theorems are (not always straight-forward) consequences.
Point of Post: In this post we discuss an interesting relationship between Sylow’s Theorems and the direct product of groups. Namely, we’ll show that all the Sylow subgroups of a group are normal if and only if the group is isomorphic to the direct product of those Sylow Subgroups.
Often it happens, finagling around with the conditions on the number of Sylow -subgroups of some group that one finds that for every prime that divides . We shall show that this is a very fortuitous thing because then will be isomorphic to the direct product of those Sylow subgroups.
Point of post: This is a continuation of this post.
Point of post: In this post we give some alternate proofs of Sylow’s theorems which use a little more machinery then the “classic” ones.
As remarked the Sylow Theorems are probably the most fundamentally important set of theorems in finite group theory. Consequently, any new proof verifying their validity is welcomed. In this post we give alternate proofs (in some sense) of the Sylow theorems than the “classic” proofs we have already given. They are less constructive and use more machinery. That said, they are very elegant and “cute”.