# Article

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Keywords:
infinite \$aS\$-group; supplemented subgroup; nilpotent group
Summary:
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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