= Minimal cyclic factorization criterion for powerful p-groups
{title2=$G=\langle a_1\rangle\cdots\langle a_d\rangle,\ d=d(G)$}
For an odd <prime number> $p$, a finite <p-group> that is a product of $d(G)$ <cyclic subgroups> is a <powerful p-group>. Indeed, $H=G/G^p$ has <exponent of a finite group> $p$ and the same minimal generator number $d$. Its cyclic factorization gives $|H|\leq p^d$, while its <Frattini quotient> has order $p^d$. Thus $\Phi(H)=1$, $H$ is an <elementary abelian group>, and $[G,G]\subseteq G^p$. The condition on the number of factors is essential: the nonabelian <Heisenberg group> over $\mathbb F_p$ has exponent $p$ and is a product of three cyclic subgroups, although its minimal generator number is two.
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