On a problem of Szép

It is proved that, if $G$ is a finite group representable as a product of an Abelian group $A$ and a group $B$, then the center of $B$ is contained in a solvable normal subgroup of $G$. The disposition of $O_p(Z(B))$ in the upper $p$-series of a finite solvable group $G$ having a factorization of this sort is determined. A corollary for locally finite groups is provided.
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