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On the product of two \(\pi\)-decomposable soluble groups. (English) Zbl 1200.20016

All groups considered in this review are finite. A group \(G\) is said to be \(\pi\)-decomposable for a set of primes \(\pi\) if \(G\) is the direct product of a \(\pi\)-subgroup \(G_\pi=O_\pi(G)\) and a \(\pi'\)-subgroup \(G_{\pi'}=O_{\pi'}(G)\), where \(\pi'\) denotes the complementary set of \(\pi\) in the set of all primes.
Of course, every nilpotent group is \(\pi\)-decomposable, but obviously the converse is not true in general. Bearing in mind the relationship between the Hall subgroups of the group and the ones of the factors in products of nilpotent groups, the authors conjecture the following: Let the group \(G=AB\) be a product of two \(\pi\)-decomposable subgroups \(A=O_\pi(A)\times O_{\pi'}(A)\) and \(B=O_\pi(B)\times O_{\pi'}(B)\) where \(\pi\) is a set of primes. Then \(O_\pi(A)O_\pi(B)=O_\pi(B)O_\pi(A)\) is a Hall \(\pi\)-subgroup of \(G\).
The main result of this paper confirms the above conjecture for products of soluble groups and a set of odd primes \(\pi\).
Theorem. Let \(\pi\) be a set of primes. If the group \(G=AB\) is the product of two \(\pi\)-decomposable subgroups \(A=A_\pi\times A_{\pi'}\) and \(B=B_\pi\times B_{\pi'}\), then \(A_\pi B_\pi=B_\pi A_\pi\) and this is a Hall \(\pi\)-subgroup of \(G\).

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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References:

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