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(C

*n*) where C

*n* is the cyclic groupof order

*n*.

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

(also, Q(G) is written as phiof G, but i don't think rhp supports greek letters...

)