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Sylow multiplicities in finite groups | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 7، شماره 2، شهریور 2018، صفحه 1-8 اصل مقاله (200.96 K) | ||
| نوع مقاله: Ischia Group Theory 2016 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2017.21482 | ||
| نویسنده | ||
| Dan Levy* | ||
| Italy | ||
| چکیده | ||
| Let $G$ be a finite group and let $\mathcal{P}=P_{1},\ldots,P_{m}$ be a sequence of Sylow $p_{i}$-subgroups of $G$, where $p_{1},\ldots,p_{m}$ are the distinct prime divisors of $\left\vert G\right\vert $. The Sylow multiplicity of $g\in G$ in $\mathcal{P}$ is the number of distinct factorizations $g=g_{1}\cdots g_{m}$ such that $g_{i}\in P_{i}$. We review properties of the solvable radical and the solvable residual of $G$ which are formulated in terms of Sylow multiplicities, and discuss some related open questions. | ||
| کلیدواژهها | ||
| Sylow sequences؛ Sylow multiplicities؛ Solvable radical؛ Solvable residual | ||
| مراجع | ||
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[10] G. Kaplan and D. Levy, The Solvable Radical of Sylow Factorizable Groups, Arch. Math., 85 (2005) 490–496.
[11] G. Kaplan and D. Levy, Multiplicities in Sylow Sequences and the Solvable Radical, Bull. Austral. Math. Soc., 78 (2008) 477–486.
[12] G. Kaplan and D. Levy, Sylow Products and the Solvable Residual, Comm. Algebra, 36 (2008) 851–857.
[13] G. Kaplan and D. Levy, Products of Sylow Subgroups and the Solvable Radical in Groups St., Andrews 2005, 2, Cambridge University Press, 2007 527–530.
[14] D. Levy, The Average Sylow Multiplicity Character and Solvability of Finite Groups, Comm. Algebra, 38 (2010) 632–644.
[15] D. Levy, The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical, Comm. Algebra, 38 (2010) 4144–4154.
[16] D. Levy, The Average Sylow Multiplicity Character and the Solvable Residual, Comm. Algebra, 41 (2013) 3090–3097.
[17] G. A. Miller. The product of two or more groups, Bull. Amer. Math. Soc., 19 (1913) 303–310.
[18] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., 74 (1968) 383–437. | ||
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