Local characterizations of periodic simple groups of finitary transformations
A subgroup \(H\) of a group \(G\) is called inert if, for every \(g\in G\), the intersection \(H\cap H^g\) has finite index in \(H\). The group \(G\) itself and any of its finite subgroups are the simplest examples of inert subgroups in \(G\). Below we call these subgroups trivial. It was proved by the author that a countable, locally finite, simple group \(G\) is isomorphic to a group of Lie-type if and only if every inert subgroup in \(G\) is trivial. This result gave rise to the following question: What is the structure of a nontrivial inert subgroup of a simple group? Clearly every proper inert subgroup of a simple group is residually finite. But this property is not the only specific feature of their structure. It was shown that for every proper inert subgroup \(H\) of a locally finite simple group \(G\), one of the following statements is true. 1) The group \(H\) is locally normal; 2) There are no finite nontrivial normal subgroups in \(H\); moreover, there exists a prime number \(p\) such that \(O_p (H)\neq 1\) and the quotient group \(H/ O_p (H)\) is locally normal; 3) There are no finite nontrivial normal subgroups in \(H\) and the locally nilpotent radical of \(H\) is trivial, while the locally solvable radical \(S(H)\) of \(H\) is not; the quotient group \(H/ S(H)\) is locally normal; 4) There are no finite nontrivial normal subgroups in \(H\), and \(S(H)=1\). In this paper we consider simple groups whose proper inert subgroups are all of types 1) or 2). To formulate the results obtained, we need the following definition: a linear transformation \(g\) of a vector space \(V\) is called finitary if \(g\) acts identically on some subspace of finite codimension in \(V\). A group \(G< GL(V)\) is said to be a group of finitary transformations (of \(V\)) if each element in \(G\) is a finitary transformation (of \(V\)). Theorem 1. If \(G\) is a countable, locally finite, simple group, then the following conditions are equivalent: 1) \(G\) is isomorphic to a group of finitary transformations; 2) there exists a prime number \(p\) such that for every proper inert subgroup \(H\) in \(G\), the quotient group \(H/ O_p (H)\) is locally normal; 3) every proper inert subgroup whose locally solvable radical is trivial is locally normal. Theorem 6. For a locally finite simple group \(G\), the following conditions are equivalent: 1) \(G\) is isomorphic to a finitary transformation group; 2) every countable, semisimple, residually finite subgroup of \(G\) contains a finite nontrivial normal subgroup; 3) every semisimple residually finite subgroup of \(G\) is locally normal.
scientific article; zbMATH DE number 738946