An antichain condition for infinite groups

Let χ be a subgroup-theoretical property. We introduce an antichain condition AC χ which forbids the existence of infinite antichains of mutually permutable non- χ subgroups whose infinite joins remain non- χ . This is a “width” analogue of the real chain condition on non- χ subgroups, and it extends the usual hierarchy of weak chain conditions (double chain condition, deviation, and 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 χ of normality, almost normality, near normality, permutability, modularity, and pronormality, we prove that a generalized radical group satisfies AC χ if and only if it satisfies RCC on non- χ subgroups; equivalently, it satisfies any of the standard weak chain conditions on non- χ subgroups. In particular, we obtain minimax-type dichotomies: either the group is minimax, or every subgroup satisfies χ . This yields characterizations in terms of Dedekind groups, quasi-Hamiltonian groups, groups with modular subgroup lattice, and 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.