Groups satisfying the double chain condition on subnormal subgroups. (English) Zbl 1352.20026
The authors study the double chain condition on subnormal subgroups, i.e. the groups admitting no infinite double sequences consisting of subnormal subgroups. The main results of the article are the following theorems.
Theorem 1. Let \(G\) be a residually finite group satisfying the double chain condition on subnormal subgroup. Then, \(G\) satisfies the maximal condition on subnormal subgroups.
Theorem 2. Let \(G\) be a radical group. Then it satisfies the double chain condition on subnormal subgroup iff one of the following conditions holds:
Theorem 1. Let \(G\) be a residually finite group satisfying the double chain condition on subnormal subgroup. Then, \(G\) satisfies the maximal condition on subnormal subgroups.
Theorem 2. Let \(G\) be a radical group. Then it satisfies the double chain condition on subnormal subgroup iff one of the following conditions holds:
- 1.
- \(G\) satisfies the minimal condition on subnormal subgroups.
- 2.
- \(G\) satisfies the maximal condition on subnormal subgroups.
- 3.
- \(G=HJ\) where \(J\) is the finite residual of \(G\), \(H\) is polycyclic, the centralizer of \(J\) in \(H\) is finite and every subnormal subgroups of \(G\) either properly contains \(J\) or is a Chernikov group.
Reviewer: Igor Subbotin (Los Angeles)
MSC:
| 20E15 | Chains and lattices of subgroups, subnormal subgroups |
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