# ON DECOMPOSABILITY OF FINITE GROUPS

• Arhrafi, Ali-Reza (Department of Mathematics Faculty of Science University of Kashan)
• Published : 2004.05.01

#### Abstract

Let G be a finite group and N be a normal subgroup of G. We denote by ncc(N) the number of conjugacy classes of N in G and N is called n-decomposable, if ncc(N) = n. Set $K_{G}\;=\;\{ncc(N)$\mid$N{\lhd}G\}$. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X. In this paper we characterise the {1, 3, 4}-decomposable finite non-perfect groups. We prove that such a group is isomorphic to Small Group (36, 9), the $9^{th}$ group of order 36 in the small group library of GAP, a metabelian group of order $2^n{2{\frac{n-1}{2}}\;-\;1)$, in which n is odd positive integer and $2{\frac{n-1}{2}}\;-\;1$ is a Mersenne prime or a metabelian group of order $2^n(2{\frac{n}{3}}\;-\;1)$, where 3$\mid$n and $2\frac{n}{3}\;-\;1$ is a Mersenne prime. Moreover, we calculate the set $K_{G}$, for some finite group G.

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#### Cited by

1. On 9- and 10-decomposable finite groups vol.26, pp.1-2, 2008, https://doi.org/10.1007/s12190-007-0001-8
2. On finite x-decomposable groups for X = {1, 2, 4} vol.53, pp.3, 2012, https://doi.org/10.1134/S0037446612020255