Proceedings Mathematical Sciences

, Volume 114, Issue 3, pp 217–224

# On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

• Ali Reza Ashrafi
• Geetha Venkataraman
Article

## Abstract

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetKG = {ncc(A)¦A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifKG =X.

Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z6, D8, Q8, S4, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].

### Keywords

Finite group n-decomposable subgroup conjugacy class X-decomposable group

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### References

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