infinite $aS$-group; supplemented subgroup; nilpotent group
Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an $aS$-group. Finally, it is shown that if $G$ is an $aS$-group and $|G|\neq pq,p$, where $p$ and $q$ are primes, then $G$ has a triple factorization.
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