Groups whose primary subgroups are normal sensitive.
A subgroup \(H\) of a group \(G\) is called \textit{normal-sensitive} if \(N=H\cap N^G\) for every normal subgroup \(N\) of \(H\). It was proved by \textit{S. Bauman} [Arch. Math. 25, 337-340 (1974; Zbl 0291.20028)] that all subgroups of a finite soluble group are normal-sensitive if and only if \(G\) is a T-group. Recall here that a group \(G\) is said to be a \textit{T-group} if normality in \(G\) is a transitive relation, i.e. if every subnormal subgroup of \(G\) is normal. In this interesting paper, the authors investigate the structure of locally finite groups whose primary subgroups are normal-sensitive. In fact, they prove that if \(G\) is a locally finite group and \(L\) is the locally nilpotent residual of \(G\), then every \(p\)-subgroup of \(G\) is normal-sensitive, for all primes \(p\in\pi(L)\), if and only if the following conditions hold: (i) \(2\notin\pi(L)\), (ii) \(L\) is Abelian, (iii) \(\pi(L)\cap\pi(G/L)=\emptyset\), (iv) every subgroup of \(L\) is normal in \(G\), (v) \([H,G]=H\) for each non-trivial subgroup \(H\) of \(L\). It follows that every primary subgroup of a locally finite group \(G\) is normal-sensitive if and only if \(G\) and all its subgroups are T-groups. The authors are also able to describe normal-sensitivity in locally finite groups in terms of \(S\)-permutability (here a subgroup \(H\) of a group \(G\) is \textit{\(S\)-permutable} if \(HS=SH\) for each Sylow subgroup \(S\) of \(G\)).
scientific article; zbMATH DE number 6363096
scientific article