Groups with small deviation for non-subnormal subgroups.
If \(A\) is a partially ordered set with ordering \(\leq\) and if \(a,b\in A\) then let \([a,b]=\{x\in A:a\leq x\leq b\}\). If \(A\neq\emptyset\) and if \(A\) has the minimal condition relative to \(\leq\) then the deviation of \(A\), \(\text{dev}(A)\) is defined to be \(0\). For a general ordinal \(\alpha\) then \(\text{dev}(A)=\alpha\) provided \(\text{dev}(A)\neq\beta<\alpha\) and in any descending chain \(a_1\geq a_2\geq\cdots\) of elements of \(A\), all but finitely many of the closed intervals \([a_{n+1},a_n]\) have deviation less than \(\alpha\). Thus \(\alpha\) is the minimal ordinal with the property that \(\text{dev}([a_{n+1},a_n])<\alpha\) for all but finitely many of the \([a_{n+1},a_n]\). When \(G\) is a group and \(S\) is a family of subgroups of \(G\) then \(S\) is partially ordered by inclusion. If \(S\) is the set of all non-subnormal subgroups of \(G\) then \(\text{dev}_{\text{non-sn}}(G)\) denotes the non-subnormal deviation of \(G\). The groups \(G\) for which \(\text{dev}_{\text{non-sn}}(G)=0\) are precisely the groups with the minimal condition on non-subnormal subgroups. The groups \(G\) for which \(\text{dev}_{\text{non-sn}}(G)=1\) include the groups with the weak minimal condition on non-subnormal subgroups, but there is an example of a group \(G\) such that \(\text{dev}_{\text{non-sn}}(G)=1\) for which \(G\) does not have the minimal condition on non-subnormal subgroups. The goal of the paper is to study those groups \(G\) for which \(\text{dev}_{\text{non-sn}}(G)\leq 1\). The main results concern soluble and locally nilpotent groups. This interesting paper contains many lovely results; for example a torsionfree locally nilpotent group \(G\) with non-subnormal deviation at most \(1\) is necessarily nilpotent.
scientific article; zbMATH DE number 5637473
scientific article