Siberian Mathematical Journal ( IF 0.7 ) Pub Date : 2021-01-29 , DOI: 10.1134/s0037446621010110 Y. Mao , X. Ma , W. Guo
We prove that \( G \) is a finite \( \sigma \)-soluble group with transitive \( \sigma \)-permutability if and only if the following hold: (i) \( G \) possesses a complete Hall \( \sigma \)-set \( {\mathcal{H}}=\{H_{1},\dots,H_{t}\} \) and a normal subgroup \( N \) with \( \sigma \)-nilpotent quotient \( G/N \) such that \( H_{i}\cap N\leq Z_{\mathfrak{U}}(H_{i}) \) for all \( i \); and (ii) every \( \sigma_{i} \)-subgroup of \( G \) is \( \tau_{\sigma} \)-permutable in \( G \) for all \( \sigma_{i}\in\sigma(N) \).
中文翻译:
有限$ \ sigma $-可溶$ P \ sigma T $-组的新特征
我们证明 \(G \)是一个有限的(\ s \ \ sigma \)-可溶基团,具有传递 \(\ sigma \)-可置换性,当且仅当以下条件成立:(i)\(G \)具有完整的霍尔\(\ sigma \) -set \({\ mathcal {H}} = \ {H_ {1},\ dots,H_ {t} \} \)和一个普通子组\(N \)与 \(\ sigma \) -nilpotent商数\(G / N \) ,使得\(H_ {I} \帽ñ\当量Z _ {\ mathfrak【U}}(H_ {I})\)对于所有 \(I \) ; (ii)\(G \)的每个\(\ sigma_ {i} \)-子组 都是\(\ tau _ {\ sigma} \)-可在\(G \)表示所有\(\ sigma_ {i} \ in \ sigma(N)\)。