Atlas: Some properties of the supersoluble formation and the supersoluble residual of a group by Hassan Naraghi

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BLAST 2008
August 6-10, 2008
University of Denver
Denver, CO, USA

Organizers
Rick Ball, Natasha Dobrinen (co-chair), Nikolaos Galatos (co-chair)

Some properties of the supersoluble formation and the supersoluble residual of a group
by
Hassan Naraghi
Islamic Azad University, Ashtian Branch, Iran.

1.9375in

Some properties of the supersoluble formation and the supersoluble residual of a group

Hassan Naraghi

Department of Mathematics, Islamic Azad University, Ashtian Branch P. O. Box 39618-13347, Ashtian, Iran.

Abstract

Let p, q, r be primes such that pq is not divisor of r-1 and p < q < r. We say that the subgroups H and K of a group G are mutually permutable if H permutes with every subgroup of K and K permutes with every subgroup of H. If G=HK and K are mutually permutable, we say that G is the mutually permutable product of the subgroups H and K.
It is known that the class U of all finite supersoluble group is a formation. This means that if a finite group G is supersoluble and N is a normal subgroup of G, then G/N is supersoluble and if M and N are two norrmal subgroups of a finite group G, then G/(M∩N) is supersoluble provided that G/M and G/N are supersoluble. Consequently, every finite group G has a smallest normal subgroup with a supersoluble quotient. This subgroup is called supersoluble residual of G and it is denoted by G^U. It is clear that G^U is epimorphism-invariant and so it is a characteristic subgroup of G(see[2;II, 2.4]).
This paper focuses attention on the study of supersoluble subgroups and supersoluble residual of the group G=[W][V]X to semidirect product and consider the subgroups H=[W]X and K=[W]V of G such that X is the cyclic group of order p, and V is an irreducible and faithful X-module over GF(q), and Y=[V]X is the corresponding semidirect product and W is an irreducible and faithful Y-module over GF(r). we determine that G is the mutually permutable product of subgroups H and K. Moreover, H is not a supersoluble subgroup of G. On the other hand, K ∈ U. Moreover H^U < W. However G^U=W.

References

[]: A. Ballester-Bolinhes, M. C. Pedraza-Aguilera, , M. D. Pe'rez-Ramps , On finite products of totally permutable group, bull. Austral. Math. Soc. 53(1996), 441-445.
[]: K. Doerk, T. Hawkes, "Finite Soluble Groups", De Gruyter, Berlin/New York, 1992.
[]: Derek J.S. Robinson, Ä course in the theory of Groups", Second Edition, Springer-Verlag, New York, Inc., 1996.
[]: Joseph J. Rotman, Än Introduction to the theory of Groups", Fourth Edition, Springer-Verlag, New York, 1995.
[]: D, W. Sharpe, P. Vamos, Ïnjective module", Cambridge, at the university press, 1972.

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Date received: May 14, 2008