Two nonsimplicity criteria for groups with an infinitely isolated subgroup
A subgroup H of a group G is infinitely isolated in G if for every nontrivial element \(h\in H\) for which \(C_ G(h)\) is infinite and \(C_ G(h)\) contains an involution, we have \(C_ G(h)\leq H\). A proper subgroup H containing an involution is strongly embedded in a supergroup G if for all \(x\in G\backslash H,\quad H\cap x^{-1}Hx\) contains no involution. It is noted that the first of these two concepts was crucial for the second author's abstract characterization of groups of the form PSL(2,K) (where K is any infinite, locally finite field of odd characteristic), and also that the situation of a strongly embedded, infinitely isolated subgroup has shown itself to be germane to the solution of Chernikov's minimum problem for locally finite groups and for other classes of periodic groups. In the present paper the authors study this situation by itself; i.e. they consider the class of periodic groups with a strongly embedded, infinitely isolated subgroup. Two theorems are proved giving criteria under which a group in this class has (among other things) a non-trivial abelian normal subgroup. The paper ends with a list of examples in particular setting limitations on possible strengthenings of the theorems and showing that the class considered is fairly wide (containing nonlocally finite groups). The paper seems to be carefully written and includes a preliminary list of the definitions and earlier results pertaining to it.
scientific article; zbMATH DE number 3851368