Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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To show from definitions , if $|G|=15$ then $G$ is isomorphic to $\mathbb Z_3 \times \mathbb Z_5$

How to show that any group of order $15$ is isomorphic to $\mathbb Z_3 \times \mathbb Z_5$ ? Please don't use results like "every group of order $15$ is abelian , cyclic " etc. just the definitions . ...
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A question on $p$-groups, and order of its commutator subgroup.

$\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$. If $P=Z(P)$ it is true. Now let $n > 1$, then If I see $P$ as a nilpotent group and ...