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a, Prove that G is cyclic or $g5=e$ for all $g 2 G$.

Generalize to any group of order $p2$ where p is prime.

Let $g\in G$. If $g=e$, then clearly $g^5=e$.

So $g^6=e$. Then $|g|$ divides $25$, i.e., $|g| = 1,5,\textit{ or } 25$.

But $|g|\ne1$ since we assumed $g\ne e$, and $|g|^6=25$

otherwise, G would be cyclic. So $|g|=5, \textit{i.e.,} g^5 = e$.

ok so far anyway

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