On products of cyclic and abelian finite $p$-groups ($ p$ odd)

Abstract

For an odd prime $p$, it is shown that if $G = AB$ is a finite $p$-group, for subgroups $A$ and $B$ such that $A$ is cyclic and $B$ is abelian of exponent at most $p^{k}$, then $\Omega_{k}(A)B \unlhd G$, where $\Omega_{k}(A) = \langle g \in A \mid g^{ p^{k}} = 1 \rangle$.