Uncountable groups in which all large subgroups are permutably complemented

Abstract

A subgroup $H$ of a group $G$ is permutably complemented if it admits a permutable complement, that is, a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K=\{1\}$. A group $G$ in which every subgroup is permutably complemented is said to be a $C$-group. The main purpose of this paper is to investigate the behaviour of uncountable groups of regular cardinality $\aleph$ in which all subgroups of cardinality $\aleph$ are permutably complemented, with the particular aim of identifying sufficient conditions for such groups to be $C$-groups. Finally, in such a context, uncountable soluble groups of cardinality $\aleph$ are considered by restricting attention to only subnormal subgroups of cardinality $\aleph$ whose subnormal defect is at most $2$.