‎$‎4‎$‎-Quasinormal subgroups of prime order

Document Type : Ischia Group Theory 2018

Author

University of Warwick

Abstract

‎Generalizing the concept of quasinormality‎, ‎a subgroup $H$ of a group $G$ is said to be 4-quasinormal in $G$ if‎, ‎for all cyclic subgroups $K$ of $G$‎, ‎$\langle H,K\rangle=HKHK$‎. ‎An intermediate concept would be 3-quasinormality‎, ‎but in finite $p$-groups‎ - ‎our main concern‎ - ‎this is equivalent to quasinormality‎. ‎Quasinormal subgroups have many interesting properties and it has been shown that some of them can be extended to 4-quasinormal subgroups‎, ‎particularly in finite‎ ‎$p$-groups‎. ‎However‎, ‎even in the smallest case‎, ‎when $H$ is a 4-quasinormal subgroup of order $p$ in a finite $p$-group $G$‎, ‎precisely how $H$ is embedded in $G$‎ ‎is not immediately obvious‎. ‎Here we consider one of these questions regarding the commutator subgroup $[H,G]$‎.

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References

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International Journal of Group Theory
Volume 9, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory 2018
March 2020
Pages 25-30

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History

  • Receive Date: 17 October 2018
  • Revise Date: 04 December 2018
  • Accept Date: 09 December 2018
  • Published Online: 01 March 2020

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