Finite groups in which the normal closures of non-normal subgroups have the same order (Q2816887)
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scientific article; zbMATH DE number 6619784
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Finite groups in which the normal closures of non-normal subgroups have the same order |
scientific article; zbMATH DE number 6619784 |
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Finite groups in which the normal closures of non-normal subgroups have the same order (English)
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26 August 2016
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Dedekind groups
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normal closure
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center of a group
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If \(H\) is a subgroup of a finite group \(G\), then its normal closure is equal to the intersection of all normal subgroups of \(G\) containing \(H\). Let \(\Delta(G)\) be the number of normal closures of nonnormal subgroups of \(G\). In this paper, the groups \(G\) satisfying \(|\Delta(G)|=1\) are classified (Theorem 2.3 for nonnilpotent groups and Theorems 3.6 and 3.7 for nilpotent groups). The proof of Theorem 3.7 is fairly long and nontrivial.
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