Quasirecognition by prime graph of ²D_n(3^α) where n = 4m + 1 ≥ 21 and α is odd

چکیده مقاله

Let G be a finite group. The prime graph of G is denoted by Γ(G). In this paper, as the main result, we show that if G is a finite group such that Γ(G) = Γ(² D_n(3^α)), where n = 4m+ 1 and α is odd, then G has a unique non-Abelian composition factor isomorphic to ² D_n(3^α). We also show that if G is a finite group satisfying {pipe}G{pipe} = {pipe}² D_n(3^α){pipe}, and Γ(G) = Γ(² D_n(3^α)), then G ≅ ² D_n(3^α). As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for ² D_n(3^α). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that ² D_n(3^α) is quasirecognizable by the spectrum. © 2014 Pleiades Publishing, Ltd.