An antichain condition for infinite 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.