On locally finite groups with a small centralizer of a four-subgroup.
In this interesting paper, the authors prove the following, see 1.1, 1.2 and 4.4. Let \(G\) be a locally finite group containing a Klein-4 subgroup \(V\) such that its centralizer \(C_G(V)\) in \(G\) is finite. If \(V\) contains an element \(v\) whose centralizer in \(G\) has finite exponent, then \(G\) is (locally soluble)-by-finite and \([G,V]\) has finite index in \(G\). If \(V\) contains distinct involutions \(v\) and \(w\) such that \(C_G(v)\) and \(C_G(w)\) both have finite exponent, then \([G,V]'\) has finite exponent. Thus in the latter case we have a quite tight possible structure for \(G\).
scientific article; zbMATH DE number 5944170
scientific article