On the prime graph of D_n(q) where q ∊ {2, 5}

Abstract

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p^′ are joined by an edge if G has an element of order pp^′. Let L = D_n(q) have disconnected prime graph and q ∊ {2, 5}. In this paper, we determine finite groups G with the same prime graph as L, i.e. Γ(G) = Γ(L).