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factorizable groups; products of subgroups; $p$-groups
In this note we study finite $p$-groups $G=AB$ admitting a factorization by an Abelian subgroup $A$ and a subgroup $B$. As a consequence of our results we prove that if $B$ contains an Abelian subgroup of index $p^{n-1}$ then $G$ has derived length at most $2n$.
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