proof that a nontrivial normal subgroup of a finite -group and the center of have nontrivial intersection
Define to act on by conjugation; that is, for , , define
Note that since . This is easily seen to be a well-defined group action.
Now, the set of invariants of under this action are
We now use elementary group theory to show that divides each term on the right, and conclude as a result that divides , so that cannot be trivial.
As is a nontrivial finite -group, it is obvious from Cauchy’s theorem that for . Since and the are subgroups of , each either is trivial or has order a power of , by Lagrange’s theorem. Since is nontrivial, its order is a nonzero power of . Since each is a proper subgroup of and has order a power of , it follows that also has order a nonzero power of .
|Title||proof that a nontrivial normal subgroup of a finite -group and the center of have nontrivial intersection|
|Date of creation||2013-03-22 14:21:07|
|Last modified on||2013-03-22 14:21:07|
|Last modified by||rm50 (10146)|