Characterizations of the simple group ${^2D_n(3)}$ by prime graph and spectrum

摘要

In this paper, we prove that there exists an infinite series of finite simple groups of Lie type with connected prime graphs which are uniquely determined by their prime graphs. More precisely, we show that every finite group G with the same prime graph as ${{}^2D_{n}(3)}$ , where n ≥ 5 is odd, is necessarily isomorphic to the group ${{}^2D_{n}(3)}$ . In fact, we give a positive answer to an open problem that arose in Zavarnitsine (Algebra Logic 45(4):220–231, 2006). As a consequence of our result, we obtain that the simple group ${{}^2D_n(3)}$ , where n is an odd number, is characterizable by its spectrum.