Beidleman, James C.; Brewster, Ben; Robinson, Derek J. S. Criteria for permutability to be transitive in finite groups. (English) Zbl 0948.20015 J. Algebra 222, No. 2, 400-412 (1999). A subgroup \(H\) of a group \(G\) is said to be permutable in \(G\) if \(HK=KH\) for all subgroups \(K\) of \(G\). The authors study \(PT\)-groups, that is finite groups \(G\) such that \(H\) permutable in \(K\) and \(K\) permutable in \(G\) imply that \(H\) is permutable in \(G\). The structure of soluble \(PT\)-groups was determined by Zacher.The authors introduce the condition \(X_p\): a finite group \(G\) satisfies \(X_p\) if and only if each subgroup of a Sylow \(p\)-subgroup \(P\) of \(G\) is permutable in the normalizer \(N_G(P)\). Using Zacher’s theorem they show that \(G\) is a soluble \(PT\)-group if and only if it satisfies \(X_p\) for all primes \(p\). For this they have to study the property \(X_p\). They show that a finite group \(G\) satisfies \(X_p\) if and only if either \(G\) is \(p\)-nilpotent and has modular Sylow \(p\)-subgroups (this latter condition is missing in the statement of Theorem B of the paper) or \(G\) has an Abelian Sylow \(p\)-subgroup \(P\) and every subgroup of \(P\) is normal in \(N_G(P)\). And if \(p\) is the smallest prime divisor of \(|G|\), then \(G\) has \(X_p\) if and only if \(G\) is \(p\)-nilpotent and has modular Sylow \(p\)-subgroups.Reviewer’s remark: The proof of this last result could be shortened considerably by using Lemma 2.3.5 in the reviewer’s book “Subgroup lattices of groups” (1994; Zbl 0843.20003). Reviewer: R.Schmidt (Kiel) Cited in 2 ReviewsCited in 33 Documents MSC: 20D40 Products of subgroups of abstract finite groups 20D30 Series and lattices of subgroups 20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure 06C05 Modular lattices, Desarguesian lattices Keywords:permutable subgroups; \(T\)-groups; soluble \(PT\)-groups; modular Sylow subgroups; products of subgroups Citations:Zbl 0843.20003 PDFBibTeX XMLCite \textit{J. C. Beidleman} et al., J. Algebra 222, No. 2, 400--412 (1999; Zbl 0948.20015) Full Text: DOI References: [1] Gaschütz, W., Gruppen, in denen das Normalteilersein transitiv ist, J. reine angew. Math., 198, 87-92 (1957) · Zbl 0077.25003 [2] Huppert, B., Zur Sylowstruktur auflösbarer Gruppen, Arch. Math., 12, 161-169 (1961) · Zbl 0102.26803 [3] Iwasawa, K., Über die endlichen Gruppen und die Verbände ihrer Untergruppen, J. Fac. Sci. Imp. Univ. Tokyo Sect. I, 4, 171-199 (1941) · Zbl 0061.02503 [4] Maier, R.; Schmid, P., The embedding of quasinormal subgroups in finite groups, Math. Z., 131, 269-272 (1973) · Zbl 0259.20017 [5] Ore, O., Contributions to the theory of groups of finite order, Duke Math. J., 5, 431-460 (1939) · JFM 65.0065.06 [6] Peng, T. A., Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc., 20, 232-234 (1969) · Zbl 0167.02302 [7] Robinson, D. J.S., A note on finite groups in which normality is transitive, Proc. Amer. Math. Soc., 19, 933-937 (1968) · Zbl 0159.31002 [8] Robinson, D. J.S., A Course in the Theory of Groups (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0496.20038 [9] Schmidt, R., Subgroup Lattices of Groups (1994), de Gruyter: de Gruyter Berlin · Zbl 0843.20003 [10] Zacher, G., I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 37, 150-154 (1964) · Zbl 0136.28302 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.