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Omegas of agemos in powerful groups | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 9، شماره 3، آذر 2020، صفحه 185-192 اصل مقاله (194.86 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2019.113217.1507 | ||
| نویسنده | ||
| James Williams* | ||
| Department of Mathematical Sciences, University of Bath, UK | ||
| چکیده | ||
| In this note we show that for any powerful $p$-group $G$, the subgroup $\Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,j\geq1$ when $p$ is an odd prime, and $i\geq1$, $j\geq2$ when $p=2$. We provide an example to show why this modification is needed in the case $p=2$. Furthermore we obtain a bound on the powerful nilpotency class of $\Omega_{i}(G^{p^{j}})$. | ||
| کلیدواژهها | ||
| powerful؛ nilpotent؛ $p$-group | ||
| مراجع | ||
|
[1] J. D. Dixon, M. P. F. Du Sautoy, A. Mann and D. Segal, Analytic Pro-P Groups, Cambridge Univ. Press, Cambridge, 2003. [2] G. A. Fern´andez-Alcober, Omega subgroups of powerful p-groups, Israel J. Math., 162 (2007) 75–79.
[3] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.1, (2018).
[4] J. Gonz´alez-S´anchez and A. Jaikin-Zapirain, On the structure of normal subgroups of potent p-groups, J. Algebra, 276 (2004) 193–209. [5] A. Lubotzky and A. Mann, Powerful p-groups. I. Finite groups, J. Algebra, 105 (1987) 484–505.
[6] A. Mann, Some questions about p-groups, J. Aust. Math. Soc, 67 (1999) 356–379.
[7] A. Mann and F. Posnick-Fradkin, Subgroups of powerful groups, Israel J. Math., 138 (2003) 19–28.
[8] G. Traustason and J. Williams, Powerfully nilpotent groups, J. Algebra, 522 (2019) 80–100. | ||
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