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Centralizers in simple locally finite groups | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 2، شماره 1، خرداد 2013، صفحه 1-10 اصل مقاله (459.6 K) | ||
| نوع مقاله: Ischia Group Theory 2012 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2013.1521 | ||
| نویسنده | ||
| Mahmut Kuzucuoğlu* | ||
| چکیده | ||
| This is a survey article on centralizers of finite subgroups in locally finite, simple groups or LFS-groups as we will call them. We mention some of the open problems about centralizers of subgroups in LFS-groups and applications of the known information about the centralizers of subgroups to the structure of the locally finite group. We also prove the following: Let $G$ be a countably infinite non-linear LFS-group with a Kegel sequence $\mathcal{K}=\{(G_i,N_i)\ |\ \ i\in \mathbf{N}\ \}$. If there exists an upper bound for $\{ |N_i| \ | \ \ i\in \mathbf{N}\ \}$, then for any finite semisimple subgroup $F$ in $G$ the subgroup $C_G(F)$ has elements of order $p_i$ for infinitely many distinct prime $p_i$. In particular $C_G(F)$ is an infinite group. This answers Hartley's question provided that there exists a bound on $\{ |N_i| \ | \ \ i\in \mathbf{N}\ \}$ | ||
| کلیدواژهها | ||
| Centralizer؛ locally finite؛ simple groups | ||
| مراجع | ||
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