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Yakov
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The Sylow subgroup $\Sigma_n$ of the Symmetric group $S_{p^n}$ has only one maximal normal abelian subgroup, say $B$, for $p>2$ and $n>1$. As $\Sigma_n/B$ is isomorphic to
$\Sigma_{n-1}$, the group $\Sigma_n$ yields a positive answers on Questions 1 and 2.

Y.B.

Yakov
  • 103
  • 3