# Frattini subgroup

1. Oct 9, 2009

### math_grl

1. The problem statement, all variables and given/known data

G is a finite p-group, show that $$G/ \Phi (G)$$ is elementary abelian p-group.

2. Relevant equations

$$\Phi (G)$$ is the intersection of all maximal subgroups of G.

3. The attempt at a solution

By sylow's theorem's we have 1 Sylow p-subgroup which is normal, call P. Then the order of G/P = p, so it's cyclic and thus abelian. Since there is only one maximal subgroup then $$\Phi (G) = P$$???.

I'm having trouble convincing myself this is complete. I'm told that $$G/\Phi (G)$$ can be of prime powered order. I'm afraid i might be missing something.