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A note on infinite \(aS\)-groups. (English) Zbl 1363.20036

Let \(G\) be a group. If every nontrivial subgroup of \(G\) has a proper supplement (complement), then \(G\) is called an \(aS\)-group (\(aC\)-group). L.-C. Kappe and J. Kirtland [Glasg. Math. J. 42, No. 1, 37–50 (2000; Zbl 0945.20016)] showed that the class of finite \(aS\)-groups coincides with the class of finite \(aC\)-groups which were characterized by P. Hall [J. Lond. Math. Soc. 12, 201–204 (1937; Zbl 0016.39301)]. In the article under review, the authors study infinite \(aS\)-groups and they have obtained some interesting characterizations of these groups. In particular, they proved that a nilpotent group \(G\) is an \(aS\)-group iff \(G\) is a subdirect product of cyclic groups of prime orders. Another new result states that if \(G\) is an \(aS\)-group satisfying the descending chain condition on subgroups, then \(G\) is finite. Also, the authors characterize all abelian groups whose every nontrivial factor group is an \(aS\)-group. Finally, the problem of a triple factorization was also investigated for \(aS\)-groups.

MSC:

20E34 General structure theorems for groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F18 Nilpotent groups
20E22 Extensions, wreath products, and other compositions of groups
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References:

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