##
**Infinite groups with short balanced chains of subgroups.**
*(English)*
Zbl 1151.20022

A subgroup \(H\) of a group \(G\) is abnormal in \(G\) if \(g\in\langle H,H^g\rangle\) for each element \(g\in G\). Abnormal subgroups are of interest because if \(H\leq K\leq G\) with \(H\) abnormal in \(G\) then \(K\) is self-normalizing. In this paper the authors obtain a number of interesting results connected to abnormality.

A group \(G\) is a \(T\)-group if normality is a transitive relation and \(G\) is a \(\overline T\)-group if every subgroup of \(G\) is a \(T\)-group. It is well-known that \(\overline T\lneqq T\). In part of this paper the authors obtain new characterizations of \(\overline T\)-groups. For example they prove that if \(G\) is a radical group then \(G\) is a \(\overline T\)-group if and only if every cyclic subgroup of \(G\) is abnormal in its normal closure.

Along the way to proving this result the authors establish numerous additional facts concerning groups all of whose cyclic subgroups are abnormal in their normal closures. For example, J. S. Rose [Math. Z. 90, 29-40 (1965; Zbl 0131.02403)] defined a balanced chain connecting a subgroup \(H\) to a group \(G\) as a chain of subgroups \(H=H_0\leq H_1\leq\cdots\leq H_{n-1}\leq H_n=G\) such that, for each \(j\), either \(H_j\triangleleft H_{j+1}\) or \(H_j\) is abnormal in \(H_{j+1}\). Thus the groups all of whose cyclic subgroups are abnormal in their normal closures are examples of groups in which the cyclic subgroups have a balanced chain of length \(2\).

Groups \(G\) all of whose subgroups are normal or abnormal (i.e. have balanced chains of length at most \(1\)) have been studied by A. Fattahi [J. Algebra 28, 15-19 (1974; Zbl 0274.20022)], I. Ya. Subbotin [Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 3(358), 86-88 (1992; Zbl 0816.20030)] and M. De Falco, L. A. Kurdachenko and I. Ya. Subbotin [Atti Semin. Mat. Fis. Univ. Modena 46, No. 2, 435-442 (1998; Zbl 0918.20017)] and the work under review represents the next step in a classification theory of groups with balanced chains.

A group \(G\) is a \(T\)-group if normality is a transitive relation and \(G\) is a \(\overline T\)-group if every subgroup of \(G\) is a \(T\)-group. It is well-known that \(\overline T\lneqq T\). In part of this paper the authors obtain new characterizations of \(\overline T\)-groups. For example they prove that if \(G\) is a radical group then \(G\) is a \(\overline T\)-group if and only if every cyclic subgroup of \(G\) is abnormal in its normal closure.

Along the way to proving this result the authors establish numerous additional facts concerning groups all of whose cyclic subgroups are abnormal in their normal closures. For example, J. S. Rose [Math. Z. 90, 29-40 (1965; Zbl 0131.02403)] defined a balanced chain connecting a subgroup \(H\) to a group \(G\) as a chain of subgroups \(H=H_0\leq H_1\leq\cdots\leq H_{n-1}\leq H_n=G\) such that, for each \(j\), either \(H_j\triangleleft H_{j+1}\) or \(H_j\) is abnormal in \(H_{j+1}\). Thus the groups all of whose cyclic subgroups are abnormal in their normal closures are examples of groups in which the cyclic subgroups have a balanced chain of length \(2\).

Groups \(G\) all of whose subgroups are normal or abnormal (i.e. have balanced chains of length at most \(1\)) have been studied by A. Fattahi [J. Algebra 28, 15-19 (1974; Zbl 0274.20022)], I. Ya. Subbotin [Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 3(358), 86-88 (1992; Zbl 0816.20030)] and M. De Falco, L. A. Kurdachenko and I. Ya. Subbotin [Atti Semin. Mat. Fis. Univ. Modena 46, No. 2, 435-442 (1998; Zbl 0918.20017)] and the work under review represents the next step in a classification theory of groups with balanced chains.

Reviewer: Martyn Dixon (Tuscaloosa)

### MSC:

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20F19 | Generalizations of solvable and nilpotent groups |

### Keywords:

abnormal subgroups; pronormal subgroups; soluble groups; balanced chains; normal closures; transitive normality
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\textit{L. A. Kurdachenko} et al., J. Algebra 319, No. 9, 3901--3917 (2008; Zbl 1151.20022)

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### References:

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