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Finite groups in which (\(S\)-)semipermutability is a transitive relation. (English) Zbl 1181.20024

Int. J. Algebra 2, No. 1-4, 143-152 (2008); Corrigendum 6, No. 13-16, 727-728 (2012).
Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called semipermutable in \(G\) if \(H\) permutes with every subgroup \(K\) of \(G\) with \(\gcd(|K|,|H|)=1\). The group \(G\) is called a BT-group if, on the set of subgroups of \(G\), being semipermutable is transitive. A subgroup \(H\) of \(G\) is called \(S\)-semipermutable in \(G\) if \(H\) permutes with every Sylow \(p\)-subgroup of \(G\) with \(\gcd(p,|H|)=1\) where \(p\) divides the order of \(G\). The group \(G\) is called an SBT-group if, on the set of subgroups of \(G\), being \(S\)-semipermutable is transitive.
Theorem 3.1 states that the following statements are equivalent: (1) \(G\) is a soluble BT-group, (2) \(G\) is a soluble SBT-group, (3) every subgroup of \(G\) of prime power order is semipermutable in \(G\), (4) every subgroup of \(G\) is semipermutable in \(G\), (5) every subgroup of \(G\) is \(S\)-semipermutable in \(G\), (6) every subgroup of \(G\) of prime power order is \(S\)-semipermutable in \(G\), (7) there exists an Abelian normal Hall subgroup \(L\) of \(G\) of odd order such that \(G/L\) is nilpotent and the elements of \(G\) induce power automorphisms in \(L\); moreover, for any two distinct primes \(p\) and \(q\) that do not divide the order of \(L\), the elements in \(G\) of \(p\)-power order commute with those of \(q\)-power order.
Theorem 4.1: If \(G\) is a minimal non-BT-group, then (a) \(G\) is soluble, (b) some Sylow subgroup of \(G\) is normal, (c) the number of prime divisors of the order of \(G\) is either 2 or 3.
The paper investigates further properties of minimal non-BT-groups.

MSC:

20D40 Products of subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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