On the norm and Wielandt series in finite groups. (Q2909070)

Given a finite group \(G\), the norm \(\text{N}(G)\) is the intersection of the normalisers of all subgroups of \(G\). It is well-known that \(\zeta(G)\leq\text{N}(G)\leq\zeta_2(G)\), where \(\zeta(G)\) and \(\zeta_2(G)\) denote the centre and the second centre of \(G\), respectively. A group \(G\) is said to be capable when there exists a group \(H\) such that \(H/\zeta(H)\) is isomorphic to \(G\).NEWLINENEWLINE In this paper, the authors study necessary and sufficient conditions for a capable group to satisfy \(\text{N}(G)=\zeta(G)\). For example, Theorem~3.12 states that if \(G\cong H/\zeta(H)\) is a capable group, then \(\text{N}(G)=\zeta(G)\) if one of the following conditions is satisfied: (1) every subgroup of \(H'\cap\zeta_2(H)\) is normal in \(H\); (2) \(\text{N}(H)=\zeta_2(H)\) and \(|\zeta(H)|\) is square-free; or (3) \(H\) is a regular \(p\)-group and \(\text{N}(H)=\zeta_2(H)\).NEWLINENEWLINE Metacyclic \(p\)-groups of odd order with the presentation NEWLINE\[NEWLINE\langle a,b\mid a^{p^{r+s+u}}=1,\;b^{p^{r+s+t}}=a^{p^{r+s}},\;[a,b]=a^{p^r}\rangleNEWLINE\]NEWLINE with \(r\geq 1\), \(u,s,t\geq 0\) and \(u\leq r\) are studied in Theorem~3.16. Here \(\text{N}(G)\neq\zeta_2(G)\), \(\text{N}(G)=\langle a^{p^s},\;b^{p^{s+u}}\rangle\) for \(t\geq u\geq 0\), \(\text{N}(G)=\langle a^{p^{s+u-t}},\;b^{-p^{s+t}}a^{p^s}\rangle\) for \(0\leq t<u\), and \(\text{N}(G)=\zeta(G)\) if and only if \(u=0\).NEWLINENEWLINE Theorem~3.18 shows the classification of \(2\)-generator \(p\)-groups with \(p>2\) and \(G'\) cyclic for which \(\text{N}(G)=\zeta(G)\), which fall into one of eight cases. Some properties related to the norm of \(p\)-groups of odd order with cyclic derived subgroup are presented in Theorem~3.21. For instance, \(\exp(G)\geq\exp({\text{N}}(G)/\zeta(G))^2\). Examples of groups which show the necessity of the hypotheses of the theorems are presented, as well as a non-Abelian \(2\)-group with derived subgroup in which the norm coincides with the second centre.