ABSTRACT
In the discussion of finite Abelian groups we saw that every Abelian group is isomorphic to the direct product of its Sylow subgroups. Of course this fails terribly for most non-Abelian groups, but amazingly enough, one can salvage a good part of this theorem. In the process we shall come up with a powerful new invariant — the number of p-Sylow subgroups. Our basic tool is a generalization of conjugacy classes. First we generalize centralizers of elements.
