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Tr. Inst. Mat., 2008, Volume 16, Number 1, Pages 4–8 (Mi timb47)  

Transitivity of Sylow permutability, the converse of Lagrange's theorem, and mutually permutable products

M. Asaada, A. Ballester-Bolinchesb, J. C. Beidlemanc, R. Esteban-Romerod

a Cairo University
b Universitat de València
c University of Kentucky
d Universidad Politécnica de Valencia

Abstract: This paper is devoted to the study of mutually permutable products of finite groups. A factorised group $G=AB$ is said to be a mutually permutable product of its factors $A$ and $B$ when each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of $\mathcal Y$-groups (groups satisfying the converse of Lagrange's theorem) and $\mathrm {SC}$-groups (groups whose chief factors are simple) are $\mathrm{SC}$-groups. Next, we show that a product of pairwise mutually permutable $\mathcal Y$-groups is supersoluble. Finally, we give a local version of the result stating that if a mutually permutable product of two groups is a $\mathrm{PST}$-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are $\mathrm{PST}$-groups.

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UDC: 512.542.63
Received: 03.01.2008
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Citation: M. Asaad, A. Ballester-Bolinches, J. C. Beidleman, R. Esteban-Romero, “Transitivity of Sylow permutability, the converse of Lagrange's theorem, and mutually permutable products”, Tr. Inst. Mat., 16:1 (2008), 4–8

Citation in format AMSBIB
\Bibitem{AsaBalBei08}
\by M.~Asaad, A.~Ballester-Bolinches, J.~C.~Beidleman, R.~Esteban-Romero
\paper Transitivity of Sylow permutability, the converse of Lagrange's theorem, and mutually permutable products
\jour Tr. Inst. Mat.
\yr 2008
\vol 16
\issue 1
\pages 4--8
\mathnet{http://mi.mathnet.ru/timb47}


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