# Berkovich's theorem on failure of abelian-to-normal replacement for subgroups of small index

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This article discusses a failure of replacement, i.e., a situation where the analogue of a valid replacement theorem fails to hold under slightly modified conditions.
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## Statement

Suppose $p$ is a prime number such that $p \ge 5$. Then, there exists a group of order $p^{2p + 1}$ that contains an abelian subgroup of order $p^{(3p - 1)/2}$ but does not contain any abelian normal subgroup of order $p^{(3p - 1)/2}$. In particular:

1. If $k = (3p - 1)/2$, there exists a finite $p$-group that has an abelian subgroup of order $p^k$ but no abelian normal subgroup of order $p^k$. Then, the collection of abelian groups of order $p^{(3p - 1)/2}$ is not a Collection of groups satisfying a weak normal replacement condition (?).
2. If $k =(p + 3)/2$, there exists a finite $p$-group that has an abelian subgroup of index $p^k$ but no abelian normal subgroup of index $p^k$.