(LaGrange) Proving/disproving that G; |G|= x^y is either cyclic or a^m=e for all a∈G

The following likely requires LaGrange's Theorem / cosets, but I am unsure how to construct the necessary counterexamples, or prove/disprove part (b). Any help/hints/tips on any of this would be very much appreciated.

We know that if G is a group of order 4 with identity e, then either G is cyclic or a^{2} = e for all a∈G.

(a) Find a counterexample to prove that this result cannot be generalized to use a base that is a natural number m other than 2. That is, disprove the following proposition:

If m is a natural number greater than 2 and G is a group of order m^{2} with identity e, then G is either cyclic or a^{m} = e for ever a∈G.

(b) Is there any condition on m that will make the proposition in (a) true? If yes, state and prove a conjecture. If no, explain why.

(c) Does the stated result generalize to other powers of 2? That is, prove or disprove the following proposition.

If G is a group of order 2^{k} for some k∈ℤ with k>2, then either G is cyclic or a^{2} = e for every a∈G.

Re: (LaGrange) Proving/disproving that G; |G|= x^y is either cyclic or a^m=e for all

G is a group of order m and m> 2, with m prime number

let $\displaystyle x \in G $ and x differ from the identity

from lagrange theorem, the order of x divide the order of G

so x order is m^2 or m

if it is m^2 then G is generated from x which means G is cyclic

then other case x order is m.

Re: (LaGrange) Proving/disproving that G; |G|= x^y is either cyclic or a^m=e for all

Thanks Amer, that helps tremendously with part (b).

I am still working to construct counter examples for (a) and (c), any advice on this?

Re: (LaGrange) Proving/disproving that G; |G|= x^y is either cyclic or a^m=e for all

the following counter-example works for both a) and c):

let G = Z8 x Z2, which is of order 16 (= 4^2 = 2^4).

consider the element (1,1) in G. this element is of order 8.

Re: (LaGrange) Proving/disproving that G; |G|= x^y is either cyclic or a^m=e for all

Thanks for that, however, I should have mentioned that we have only worked with Z_x, Dihedral, and U groups. Are there any counterexamples within these groups?

Re: (LaGrange) Proving/disproving that G; |G|= x^y is either cyclic or a^m=e for all