# group theory problem

genius
Posers and Puzzles 10 May '07 08:28
1. genius
Wayward Soul
10 May '07 08:282 edits
it's a problem in a past paper for one of my modules and i can't seem to work out part ii. to start with i thought it was true, then showed it was probably false but i can't seem to find a counter example which would mean my first hypothesis was right...anyway, the question:

Let G be a finite group and let x be in the group G. We define x to be a non-generator if, whenever G=< X, x>, then G=< X >.

Let Q(G) be the set of all non-generators of G.

i) Prove that Q(G) is a subgroup of G.

ii) Is it true that Q(G) is always normal in G? Justify your answer.

A subgroup H of a group G is said to be maximal if H< >G and if K< =G with H< =K< >G then H=K.

iii) Prove that Q(G) is equal to the intersection of the maximal subgroups of G.

iv) Find Q(Cn) where Cn is the cyclic groupof order n.

if you have any questions about definitions or anything, just ask! ðŸ™‚

(also, Q(G) is written as phiof G, but i don't think rhp supports greek letters...ðŸ˜›)
2. 11 May '07 11:37
Using the well-known technique of "proof by Wikipedia," I have determined that you were right the first time. It's a normal (characteristic, in fact) subgroup called the Frattini subgroup. I doubt the proof is hard -- I'll post if I come up with it.