A New Characterization of A₁₀ by its Noncommuting Graph

Abstract

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture.

Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈ ℕ | G has a conjugacy class of size n}.

In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows.

AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M.

In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A ₁₀), then G ≅ A ₁₀, where A ₁₀ is the alternating group of degree 10.

Key Words:

• Conjugacy class
• Finite group
• Noncommuting graph

2000 Mathematics Subject Classification:

• 20D05
• 20D06
• 20D60

ACKNOWLEDGMENT

The authors would like to thank the referee for his/her help.

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

Notes

Communicated by A. Yu. Olshanskii.