Berkovich's theorem on failure of abelian-to-normal replacement for subgroups of small index
From Groupprops
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.
View other failures of replacement | View replacement theorems
Statement
Suppose is a prime number such that . Then, there exists a group of order that contains an abelian subgroup of order but does not contain any abelian normal subgroup of order . In particular:
- If , there exists a finite -group that has an abelian subgroup of order but no abelian normal subgroup of order . Then, the collection of abelian groups of order is not a Collection of groups satisfying a weak normal replacement condition (?).
- If , there exists a finite -group that has an abelian subgroup of index but no abelian normal subgroup of index .
Related facts
- Abelian-to-normal replacement theorem for prime-square index
- Jonah-Konvisser congruence condition on number of abelian subgroups of prime-square index for odd prime
- Abelian-to-normal replacement theorem for prime-cube index for odd prime
- Alperin's conjecture on abelian-to-normal replacement for small index
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 copy}^{More info}, Page 540 (Page 69 within the document), Section 14: On Alperin's conjecture on abelian subgroups of small index