Research articles
ScienceAsia (): 125128 doi:
10.2306/scienceasia15131874...125
OVDcharacterization of simple K_{3}groups
Shitian’Liu^{a,b}, Donglin’Lei^{a}, Xianhua’Li^{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=p_{1}^{α1}p_{2}^{α2}...p_{k}^{αk}p_{k+1}^{αk+1}...p_{n}^{αn}, where the p_{i} are different primes and the α_{i} are positive integers. Assume that π_{V}(G)={p_{1},p_{2},...,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(p_{1}),deg(p_{2}),...,deg(p_{k})), where p_{1}<p_{2}<...<p_{k}. We call VD(G) the vanishing degree pattern of G. In this paper, we give a characterization of simple K_{3}groups by group orders and their vanishing degree patterns of the vanishing prime graphs.
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^{a} 
School’of’Mathematical’Science, Soochow’University, Suzhou, Jiangsu, 251125, China 
^{b} 
School’of’Science, Sichuan’University’of’Science’and’Engineering, Zigong’Sichuan, 643000, China 
* Corresponding author, Email: xhli@suda.edu.cn
Received 27 Jun 2017, Accepted 1 Jan 2018
