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$4$-Quasinormal subgroups of prime order | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 9، شماره 1، خرداد 2020، صفحه 25-30 اصل مقاله (187.53 K) | ||
| نوع مقاله: Ischia Group Theory 2018 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2018.113482.1510 | ||
| نویسنده | ||
| Stewart Edward Stonehewer* | ||
| University of Warwick | ||
| چکیده | ||
| 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 $[H,G]$. | ||
| کلیدواژه‌ها | ||
| Finite group؛ Sylow subgroup؛ abnormal subgroup؛ seminormal subgroup | ||
| مراجع | ||
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[1] J. Cossey and S. E. Stonehewer, Generalizing Quasinormality, Int. J. Group Theory, 4 (2015) 33-39.
[2] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p Groups, Cambridge University Press, 1991.
[3] B. Huppert, Endliche Gruppen, 1, Springer-Verlag, Berlin Heidelberg New York, 1967.
[4] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer-Verlag, Berlin Heidelberg New York, 1972. [5] E. Schenkman, Group Theory, D. Van Nostrand Co., 1965.
[6] S. E. Stonehewer, Generalized Quasinormal Subgroups of Order p2, Adv. Group Theory Appl., 1 (2016) 139-149. | ||
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