A characterization of L 3(4) by its character degree graph and order.

Shitian Liu,Yunxia Xie

doi:10.1186/s40064-016-1785-5

Abstract

Let G be a finite group. The character degree graph Gamma (G) of G is the graph whose vertices are the prime divisors of character degrees of G and two vertices p and q are joined by an edge if pq divides the character degree of G. Let L_n(q) be the projective special linear group of degree n over a finite field of order q. Khosravi et. al. have shown that the simple groups L_2(p^2), and L_2(p) where pin {7,8,11,13,17,19} are characterizable by the degree graphs and their orders. In this paper, we give a characterization of L_3(4) by using the character degree graph and its order.

Highlights

In this paper all groups are finite
The graph Ŵ(G) is called character degree graph whose vertices are the prime divisors of character degrees of the group G and two vertices p and q are joined by an edge if pq divides some character degree of G (Manz et al 1988)
G has a normal series 1 H K G, such that K/H is a direct product of isomorphic non-abelian simple groups and |G/K | | |Out(K /H )|

Summary

Background

Let G be a finite group and let Irr(G) be the set of all irreducible characters of G. Khosravi et al (2015) proved that the group L2(p2), where p is a prime, is characterizable by its character degree graph and its order. Let Ln(q) be the special linear group of degree n over finite field of order q. Lemma 1 (Isaacs 1994, Theorem 6.5) Let A G be abelian. We need the structure of non-abelian simple group whose largest prime divisor is less than 7.

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Conclusion