A New Characterization of A10 by its Noncommuting Graph
Titel:
A New Characterization of A10 by its Noncommuting Graph
Auteur:
Wang, Lingli Shi, Wujie
Verschenen in:
Communications in algebra
Paginering:
Jaargang 36 (2008) nr. 2 pagina's 523-528
Jaar:
2008-02
Inhoud:
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 ∈ | 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) ≅ ∇ (A10), then G ≅ A10, where A10 is the alternating group of degree 10.
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