Imperfect groups
A group is imperfect if all its perfect quotient groups are trivial. The authors begin the study of these groups by establishing closure properties of these groups. In particular, an arbitrary group \(H\) is isomorphic with a 2-step subnormal subgroup of an imperfect group \(G\) such that \(| G|= 2| H|^ 2\). Here is given a criterion for a group \(H\) to be isomorphic with a normal subgroup of an imperfect group. For a non-abelian simple group \(H\) this criterion means that \(\text{Out }H\neq 1\). There is given a construction of a countably infinite, locally finite group with minimal condition on subnormal subgroups, which is a union of a chain of imperfect normal subgroups. The imperfect radical and residual of the groups \(GL_ n(\mathbb{Z})\) and \(SL_ n(\mathbb{Z})\) is calculated. Let \(R\) be a ring such that \(sr(R)=1\). If \(n\geq 1\), then \(GL_ n(R)\) is imperfect if and only if \(GL_ n(R/M)\) is imperfect for any maximal ideal \(M\) of \(R\). In particular, let \(R\) be a commutative semilocal ring which does not have the field \(\mathbb{F}_ 2\) as a quotient ring. If \(n>1\), then \(GL_ n(R)\) is imperfect if and only if \(F^*\neq(F^*)^ n\) for every quotient field \(F\) of \(R\).
minimal condition on subnormal subgroups
scientific article; zbMATH DE number 427653
scientific article