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