Groups in which all subgroups are pronormal.
A subgroup \(H\) in a group \(G\) is pronormal if for any element \(g\) in \(G\) there exists an element \(c\in N(H)\) such that \(gc\) belongs to the subgroup generated by \(H\) and \(gHg^{-1}\). Let \(G\) be a group in which any subgroup is pronormal. If \(G\) is locally solvable and non-periodic, then \(G\) is Abelian. Suppose any finitely generated subgroup \(H\neq 1\) has a subgroup \(H_ 1\) of finite index \(r>1\) and \(G\) is periodic. Then \(G\) is a semidirect product of an Abelian \(2'\)-group \(A\) and a Dedekind group \(B\). Each subgroup in \(A\) is normal in \(G\); \(\pi(B)\cap\pi(A)\) is empty; \(G'=A\times B'\); \(G\) is a semidirect product of \(A\) and any \(\pi(B)\)-subgroup in \(G\). This theorem extends results of \textit{T. A. Peng} [Proc. Am. Math. Soc. 20, 232-234 (1969; Zbl 0167.02302)] and \textit{A. Fattahi} [J. Algebra 28, 15-19 (1974; Zbl 0274.20022)].
scientific article; zbMATH DE number 4046011
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