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.