# Normal subgroups of finite solvable groups

Let $G$ be a finite solvable group, $N$ a nontrivial abelian normal subgroup of prime exponent $p$. Let $Q$ be a $p$-Sylow subgroup of $G$ containing $N$.

Is it possible that the normal core of $Z(Q)$ in $G$ is trivial?

## migrated from mathoverflow.netFeb 23 '14 at 6:19

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Yes, that's possible. Let $G=S_4$ be the symmetric group on $4$ letters, and $Q$ be a Sylow $2$-subgroup. Then the core of $Q$ in $G$ is elementary abelian of order $4$. On the other hand, $Z(Q)$ has order $2$, hence its core is trivial because $S_4$ has a trivial center.