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.

Related facts

References

=Journal references

  • On subgroups and epimorphic images of finite p-groups by Yakov Berkovich, Journal of Algebra, ISSN 00218693, Volume 248,Number 2, Page 472 - 553(Year 2002): Official copyMore info, Page 540 (Page 69 within the document), Section 14: On Alperin's conjecture on abelian subgroups of small index