ON DECOMPOSABILITY OF FINITE GROUPS

Title & Authors
ON DECOMPOSABILITY OF FINITE GROUPS
Arhrafi, Ali-Reza;

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 $K_{G}\;=\;\{ncc(N)$\small{\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 $\small{9^{th}}$ group of order 36 in the small group library of GAP, a metabelian group of order $\small{2^n{2{\frac{n-1}{2}}\;-\;1)}$, in which n is odd positive integer and $\small{2{\frac{n-1}{2}}\;-\;1}$ is a Mersenne prime or a metabelian group of order $\small{2^n(2{\frac{n}{3}}\;-\;1)}$, where 3$\small{\mid}$n and $\small{2\frac{n}{3}\;-\;1}$ is a Mersenne prime. Moreover, we calculate the set $\small{K_{G}}$, for some finite group G.
Keywords
finite group;n-decomposable subgroup;conjugacy class;
Language
English
Cited by
1.
On 9- and 10-decomposable finite groups, Journal of Applied Mathematics and Computing, 2008, 26, 1-2, 169
2.
On finite x-decomposable groups for X = {1, 2, 4}, Siberian Mathematical Journal, 2012, 53, 3, 444
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