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

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

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

iv) Find Q(C

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˜›)

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! ðŸ™‚

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