##
**Pronormality, contranormality and generalized nilpotency in infinite groups.**
*(English)*
Zbl 1070.20041

A subgroup \(H\) of a group \(G\) is called ‘pronormal’ in \(G\), if \(H\) and \(H^g\) are conjugate in \(\langle H,H^g\rangle\) for each element \(g\in G\); a subgroup \(H\) of a group \(G\) is called ‘abnormal’ in \(G\), if \(g\in\langle H,H^g\rangle\) for each element \(g\in G\); and a subgroup \(H\) of a group \(G\) is called ‘contranormal’ in \(G\) if \(H^G=G\), where \(H^G\) is the normal closure of \(H\) in \(G\). There are results involving the circumstance of the existence of such subgroups in a finite group which characterize the property of being nilpotent.

The authors of the paper under review prove some criteria of generalized nilpotency involving pronormality and abnormality. The set of elements of finite conjugacy classes in a group \(G\) forms a characteristic subgroup \(\text{FC}(G)\) called FC-center of \(G\) and a group is called an FC-group if it coincides with its FC-center. Starting from the FC-center, one can construct the ‘upper FC-central series’ of a group \(G\): \[ \langle 1\rangle=C_0\leq C_1\leq\cdots\leq C_\alpha\leq C_{\alpha+1}\leq\cdots\leq C_\gamma \] where \(C_1=\text{FC}(G)\), \(C_{\alpha+1}/C_\alpha=\text{FC}(G/C_\alpha)\), \(\alpha<\gamma\), and \(\text{FC}(G/C_\gamma)=\langle 1\rangle\). The last term \(C_\gamma\) of this series is called the upper FC-hypercenter. If \(C_\gamma=G\), then the group \(G\) is called FC-hypercentral; if \(\gamma\) is finite, then \(G\) is called FC-nilpotent. For a subgroup \(H\) of a group \(G\), one can construct the normal closure series in the following way: Put \(H_1=H^G\). If for an ordinal \(\alpha\) the subgroups \(H_\beta\) have been already constructed for all \(\beta<\alpha\), then put \(H_\alpha=\bigcap_{\beta<\alpha}H_\beta\) whenever \(\alpha\) is a limit ordinal. If \(\alpha-1\) exists and \(K=H_{\alpha-1}\), then put \(H_\alpha=H^K\). There is an ordinal \(\gamma\) such that \(H_\gamma=H_{\gamma+1}\). In other words, \(H^{H_\gamma}=H_\gamma\). A subgroup \(H\) is called ‘descending’, if the last term of the normal closure series coincides with \(H\).

The main results of the paper are:

Theorem A. Let \(G\) be a group every subgroup of which is descending. If \(G\) is FC-hypercentral, then \(G\) is hypercentral.

Theorem B. Let \(G\) be a group, \(A\) a normal Abelian subgroup of \(G\) such that \(A\leq\text{FC}(G)\) and \(G/A\) is an FC-group.

(1) If every pronormal subgroup of \(G\) is normal, then \(G\) is hypercentral.

(2) If \(G\) does not contain proper contranormal subgroups, then \(G\) is hypercentral.

(3) If \(G\) does not contain proper abnormal subgroups, then \(G\) is hypercentral.

Theorem C. Let \(G\) be a periodic group, the FC-center of which contains a \(G\)-invariant subgroup \(H\) such that \(G/H\) is a hypercentral group.

(1) If every pronormal subgroup of \(G\) is normal, then \(G\) is hypercentral.

(2) If \(G\) does not contain proper contranormal subgroups, then \(G\) is hypercentral.

(3) If \(G\) does not contain proper abnormal subgroups, then \(G\) is hypercentral.

Theorem D. Let a group \(G\) contain a Dedekind normal subgroup \(H\) such that \(G/H\) has no proper abnormal subgroup. Then abnormality is a transitive relation in \(G\).

The authors of the paper under review prove some criteria of generalized nilpotency involving pronormality and abnormality. The set of elements of finite conjugacy classes in a group \(G\) forms a characteristic subgroup \(\text{FC}(G)\) called FC-center of \(G\) and a group is called an FC-group if it coincides with its FC-center. Starting from the FC-center, one can construct the ‘upper FC-central series’ of a group \(G\): \[ \langle 1\rangle=C_0\leq C_1\leq\cdots\leq C_\alpha\leq C_{\alpha+1}\leq\cdots\leq C_\gamma \] where \(C_1=\text{FC}(G)\), \(C_{\alpha+1}/C_\alpha=\text{FC}(G/C_\alpha)\), \(\alpha<\gamma\), and \(\text{FC}(G/C_\gamma)=\langle 1\rangle\). The last term \(C_\gamma\) of this series is called the upper FC-hypercenter. If \(C_\gamma=G\), then the group \(G\) is called FC-hypercentral; if \(\gamma\) is finite, then \(G\) is called FC-nilpotent. For a subgroup \(H\) of a group \(G\), one can construct the normal closure series in the following way: Put \(H_1=H^G\). If for an ordinal \(\alpha\) the subgroups \(H_\beta\) have been already constructed for all \(\beta<\alpha\), then put \(H_\alpha=\bigcap_{\beta<\alpha}H_\beta\) whenever \(\alpha\) is a limit ordinal. If \(\alpha-1\) exists and \(K=H_{\alpha-1}\), then put \(H_\alpha=H^K\). There is an ordinal \(\gamma\) such that \(H_\gamma=H_{\gamma+1}\). In other words, \(H^{H_\gamma}=H_\gamma\). A subgroup \(H\) is called ‘descending’, if the last term of the normal closure series coincides with \(H\).

The main results of the paper are:

Theorem A. Let \(G\) be a group every subgroup of which is descending. If \(G\) is FC-hypercentral, then \(G\) is hypercentral.

Theorem B. Let \(G\) be a group, \(A\) a normal Abelian subgroup of \(G\) such that \(A\leq\text{FC}(G)\) and \(G/A\) is an FC-group.

(1) If every pronormal subgroup of \(G\) is normal, then \(G\) is hypercentral.

(2) If \(G\) does not contain proper contranormal subgroups, then \(G\) is hypercentral.

(3) If \(G\) does not contain proper abnormal subgroups, then \(G\) is hypercentral.

Theorem C. Let \(G\) be a periodic group, the FC-center of which contains a \(G\)-invariant subgroup \(H\) such that \(G/H\) is a hypercentral group.

(1) If every pronormal subgroup of \(G\) is normal, then \(G\) is hypercentral.

(2) If \(G\) does not contain proper contranormal subgroups, then \(G\) is hypercentral.

(3) If \(G\) does not contain proper abnormal subgroups, then \(G\) is hypercentral.

Theorem D. Let a group \(G\) contain a Dedekind normal subgroup \(H\) such that \(G/H\) has no proper abnormal subgroup. Then abnormality is a transitive relation in \(G\).

Reviewer: Alireza Abdollahi (Isfahan)

### MSC:

20F19 | Generalizations of solvable and nilpotent groups |

20F24 | FC-groups and their generalizations |

20E34 | General structure theorems for groups |

20F14 | Derived series, central series, and generalizations for groups |

20E07 | Subgroup theorems; subgroup growth |

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