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OVD-characterization of simple K₃-groups

ShitianÿLiu^a,b, DonglinÿLei^a, XianhuaÿLi^a,*

 
ABSTRACT:     A vanishing element of G is an element g∈G such that χ(g)=0 for some χ∈Irr(G). Let Van(G) denote the set of vanishing elements of G, i.e., Van(G)={g∈G | χ(g)=0forsomeχ∈Irr(G)}. We define vo(G) to be the set {o(g) | g∈Van(G)} consisting of the orders of the elements in Van(G), that is, vo(G)={o(g) | g∈Van(G)}. Obviously, vo(G)⊆ω(G) where ω(G) is the set of element orders of G. Let π_V(G) be the set of prime divisors of the orders of the vanishing elements of G, that is, π_V(G)={π(o(g)) | g∈Van(G)}. Obviously π_V(G)⊆π(G) where π(G) denotes the set of the prime divisors of the order |G| of a group G. Let G be a finite group and |G|=p₁^α₁p₂^α₂...p_k^α_kp_k+1^α_k+1...p_n^α_n, where the p_i are different primes and the α_i are positive integers. Assume that π_V(G)={p₁,p₂,...,p_k}. For p∈π_V(G), let deg(p):=|{q∈π_V(G)|p∼q}|, which we call the vanishing degree of p. We also define VD(G):=(deg(p₁),deg(p₂),...,deg(p_k)), where p₁<p₂<...<p_k. We call VD(G) the vanishing degree pattern of G. In this paper, we give a characterization of simple K₃-groups by group orders and their vanishing degree patterns of the vanishing prime graphs.

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* Corresponding author, E-mail: xhli@suda.edu.cn

Received 27 Jun 2017, Accepted 1 Jan 2018