Centralizers of semisimple subgroups in locally finite simple groups (Q1891520)

The locally finite simple groups (LFS-groups) are usually studied in two classes: infinite linear LFS-groups and infinite nonlinear LFS-groups. The author of this paper is mainly interested in nonlinear LFS-groups. Definition 1. Let \(G\) be a countably infinite LFS-group and \(F\) be a finite subgroup of \(G\). The group \(F\) is called a \(K\)-semisimple subgroup of \(G\), if \(G\) has a Kegel sequence \(K= (G_i, M_i)_{i\in N}\) such that \((|M_i|, |F|)=1\), \(M_i\) soluble for all \(i\) and if \(G_i/ M_i\) is a linear group over a field of characteristic \(p_i\), then \((p_i,|F|)=1\). Definition 2. A finite abelian group \(F\) in a finite simple group \(G\) of classical or alternating type is called a nice group if whenever \(G\) is of type \(B_l\) or \(D_l\), then a Sylow 2-subgroup of \(F\) is cyclic. If \(G\) is alternating or of type \(A_l\) or \(C_l\), then every abelian subgroup is a nice group. In particular every abelian group of odd order is a nice group. A finite abelian group in a countably infinite LFS-group \(G\) is called a \(K\)-nice group if \(F\) is a nice group in almost all Kegel components of a Kegel sequence \(K\) of \(G\). The main results (theorems 1 and 2) are concerned with the following question of \textit{B. Hartley} [J. Aust. Math. Soc., Ser. A 49, No. 3, 502-513 (1990; Zbl 0728.20034)]: Is it the case that in a nonlinear LFS-group the centralizer of every finite subgroup is infinite? Does the centralizer of every finite subgroup involve an infinite nonlinear simple group? Theorem 1. If \(F\) is a \(K\)-nice abelian subgroup and \(K\)-semisimple in a nonlinear LFS-group \(G\), then \(C_G (F)\) has a series of finite length in which the factors are either non-abelian simple or locally soluble moreover one of the factors is nonlinear simple. In particular \(C_G (F)\) is an infinite group. Theorem 2. Suppose that \(G\) is infinite nonlinear and every finite set of elements of \(G\) lies in a finite simple group. Then (i) There exist infinitely many abelian subgroups \(F\) of \(G\) and a locally system \(L\) of \(G\) consisting of simple subgroups such that \(F\) is nice in every member of \(L\). (ii) There exists a function \(f\) from natural numbers to natural numbers independent of \(G\) such that \(C= C_G (F)\) has a series of finite length in which at most \(f(|F|)\) factors are simple non-abelian groups for any \(F\) as in (i). Furthermore \(C\) involves a nonlinear simple group.