On finite \(T\)-groups and the Wielandt subgroup. (Q2882814)

All groups considered in the paper under review are finite. The `Wielandt subgroup' \(\omega(G)\) of a group \(G\) is the intersection of the normalisers of all subnormal subgroups of \(G\). Groups with \(\omega(G)=G\) are the groups in which every subnormal subgroup is normal, or, in other words, groups in which normality is transitive. These groups are known as `T-groups'.NEWLINENEWLINE In this paper, another Wielandt-type subgroup is defined. For a group \(G\), let \(\omega^*(G)\) be the set of all elements \(x\in G\) such that if \(x\in N_G(H)\), then \(x\in N_G(K)\) for each normal subgroup \(K\) of a subgroup \(H\) of \(G\). It is clear that \(\omega^*(G)\) is a subset of \(\omega(G)\). Moreover, \(\omega^*(G)\) is composed of all elements \(x\) of \(G\) such that \(x\in\omega(H)\) for each subgroup \(H\leq G\) such that \(x\in H\) (Lemma~2.1). Soluble T-groups are the groups such that \(\omega^*(G)=G\) (Lemma~2.2) and, for a metanilpotent group \(G\), \(\omega(G)=\omega^*(G)\). This allows the author to find some properties of the Wielandt subgroup for metanilpotent groups. For instance, if \(P\) is a Sylow \(p\)-subgroup of the metanilpotent group \(G\), \(p\) a prime, and \(P_0\leq P\), where \(P_0\) is a Sylow \(p\)-subgroup of \(\omega(G)\), then \(P_0\leq Z_2(P)\), and either \(\omega(G)\) is \(p\)-nilpotent, or \(P_0\) is a maximal Abelian subgroup of \(P\); furthermore, in the latter case \(P\) has class number at most~\(3\), and it is metabelian (Theorem~2).NEWLINENEWLINE Given a group \(G\), an element \(x\in G\) is said to be `special' when there exists a proper subgroup \(H\) of \(G\) such that \(H\) is normalised by \(x\) and \(x\) acts on \(H\) as a special automorphism (in the sense \(H^x=H\) and there exists a proper subgroup \(A\) of \(H\) such that \(H=AA^x\)). An element \(x\in G\) is `weakly special' when there exists a proper subgroup \(H\) of \(G\) such that \(H=AA^x\) for a certain \(A<H\). The set of special and weakly special elements of \(G\) is denoted by \(\text{Sp}(G)\) and \(\text{WSp}(G)\), respectively. It is clear that the special elements of \(G\) are weakly special, and it is shown that in the symmetric group of degree~\(5\) this inclusion is strict. However, in Theorem~3, it is proved that for a soluble group, \(\text{WSp}(G)=\text{Sp}(G)\) and that \(\omega^*(G)=G\setminus\text{WSp}(G)=G\setminus\text{Sp}(G)\).NEWLINENEWLINE For a group \(G\), a \(\varphi\)-pair of \(G\) is defined as a pair \((x,H)\) such that whenever \(K\), \(K^x\) are maximal subgroups of \(H\), then \(K\) and \(K^x\) are conjugate in \(H\). Theorem~5 shows that for a soluble group, \(\omega^*(G)\) is the set of all \(x\in G\) such that \((x,H)\) is a \(\varphi\)-pair whenever \(H\leq G\) and \(x\) normalises \(H\).NEWLINENEWLINE Finally, for a group with a subgroup \(K\), we say that \(K\) is a `cr-subgroup' of \(G\) if there is no \(A<K\) and \(g\in G\) such that \(K=AA^g\), and that \(K\) is a `\(\varphi\)-subgroup' of \(G\) if, for all maximal subgroups \(H\) and \(L\) of \(K\), if \(H\) and \(L\) are conjugate in \(G\), then \(H\) and \(L\) are conjugate in \(K\). With these concepts, the author characterises in Theorem~7 soluble T-groups as the groups in which \(\text{WSp}(G)=\emptyset\), as the groups with \(\text{Sp}(G)=\emptyset\), as the groups with all subgroups (all subgroups of prime power order) cr-subgroups of \(G\), and as the groups with all subgroups (all subgroups of prime power order) \(\varphi\)-subgroups of \(G\). (Also submitted to MR.)