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On 9- and 10-decomposable finite groups

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Abstract

A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. In some research papers, the question of finding all positive integer n such that there is an n-decomposable finite group was posed.

In this paper, we investigate the structure of 9- and 10-decomposable non-perfect finite groups. We prove that a non-perfect group G is 9-decomposable if and only if G is isomorphic to Aut(PSL(2,32)), Aut(PSL(3,3)), the semi-direct product Z 3 ∝(Z 5×Z 5) or a non-abelian group of order pq, where p and q are primes and p−1=8q, and also, a non-perfect finite group G is 10-decomposable if and only if G is isomorphic to Aut(PSL(2,17)), PSL(2,25):23, a split extension of PSL(2,25) by Z 2 in ATLAS notation (Conway et al., Atlas of Finite Groups, [1985]), Aut(U 3(3)) or D 38, where D 38 denotes the dihedral group of order 38.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Kashan, Kashan, 87317-51167, Iran

    Ali Reza Ashrafi

  2. Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran

    Ali Reza Ashrafi

  3. School of Mathematics, Suzhou University, Suzhou, Jiangsu, 215006, China

    Wujie Shi

Authors
  1. Ali Reza Ashrafi
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  2. Wujie Shi
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Corresponding author

Correspondence to Ali Reza Ashrafi.

Additional information

A.R. Ashrafi’s research was in part supported by a grant from IPM (No. 84200014).

W. Shi supported by the NNSF of China (Grant No. 10571128) and the SRFDP of China (Grant No. 20060285002).

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Ashrafi, A.R., Shi, W. On 9- and 10-decomposable finite groups. J. Appl. Math. Comput. 26, 169–182 (2008). https://doi.org/10.1007/s12190-007-0001-8

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  • DOI: https://doi.org/10.1007/s12190-007-0001-8

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