Centralizers of finite $p$-subgroups in simple locally finite groups

Mahmut Kuzucuoğlu

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 10 (2017) no. 3, pp. 281-286

Résumé

We are interested in the following questions of B. Hartley: (1) Is it true that, in an infinite, simple locally finite group, if the centralizer of a finite subgroup is linear, then $G$ is linear? (2) For a finite subgroup $F$ of a non-linear simple locally finite group is the order $|CG(F)|$ infinite? We prove the following: Let $G$ be a non-linear simple locally finite group which has a Kegel sequence $\mathcal{K}=\{(G_{i},1): \; i \in \mathbf{N} \}$ consisting of finite simple subgroups. Let $p$ be a fixed prime and $s\in \mathbf{N}$. Then for any finite $p-$subgroup $F$ of $G$, the centralizer $C_{G}(F)$ contains subgroups isomorphic to the homomorphic images of $SL(s,\mathbf{F}_q)$. In particular $C_G(F)$ is a non-linear group. We also show that if $F$ is a finite $p$-subgroup of the infinite locally finite simple group $G$ of classical type and given $s\in \mathbf{N}$ and the rank of $G$ is sufficiently large with respect to $|F|$ and $s$, then $C_G(F)$ contains subgroups which are isomorphic to homomorphic images of $SL(s,K)$.

Détail

Keywords: centralizer, simple locally finite, non-linear group.