An antichain condition for infinite groups

TL;DR

This paper introduces an antichain condition for infinite groups that extends existing chain conditions, showing that in generalized radical groups, it aligns with traditional hierarchy conditions and leads to minimax dichotomies.

Contribution

It defines the antichain condition c_hi and proves its equivalence to weak chain conditions in generalized radical groups, extending the hierarchy of subgroup conditions.

Findings

Antichain condition c_hi is equivalent to c on non-hi subgroups in generalized radical groups.

Groups satisfying c_hi are either minimax or have all subgroups satisfying hi.

Characterizations include Dedekind, quasi-Hamiltonian, and ar{T}-groups.

Abstract

Let $\chi$ be a subgroup-theoretical property. We introduce an \emph{antichain condition} $\operatorname{ac}_\chi$ which forbids the existence of infinite antichains of mutually permutable non-$\chi$ subgroups whose infinite joins remain non-$\chi$. This is a ''width'' analogue of the real chain condition on non-$\chi$ subgroups, and it extends the usual hierarchy of weak chain conditions (double chain condition, deviation, and $\operatorname{RCC}$). Our main results show that, within the universe of generalized radical groups, the antichain condition is as rigid as the corresponding chain conditions. For the properties $\chi$ of normality, almost normality, near normality, permutability, modularity, and pronormality, we prove that a generalized radical group satisfies $\operatorname{ac}_\chi$ if and only if it satisfies $\operatorname{RCC}$ on non-$\chi$ subgroups; equivalently, it satisfies any of the standard weak chain conditions on non-$\chi$ subgroups. In particular, we obtain minimax-type dichotomies: either the group is minimax, or \emph{every} subgroup satisfies $\chi$. This yields characterizations in terms of Dedekind groups, quasi-Hamiltonian groups, groups with modular subgroup lattice, and $\overline{T}$-groups. In the pronormal case, one has to deal with locally finite simple groups and a use of the Classification of Finite Simple Groups seems unavoidable.