od-characterization of s4(4) and its group of automorphisms

Abstract. let g be a finite group and π(g) be the set of all prime divisors of |g|. the prime graph of g is a simple graph γ(g) with vertex set π(g) and two distinct vertices p and q in π(g) are adjacent by an edge if an only if g has an element of order pq. in thiscase, we write p ∼ q. let |g| = pα1 , where p1 < p2 < . . . < pk are primesfor p ∈ π(g), let deg(p) = |{q ∈ π(g)|p ∼ q}| be the degree of p in the graph γ(g), we define d(g) = (deg(p1), deg(p2), . . . , deg(pk)) and call it the degree pattern of g. a group g is called k-fold od-characterizable if there exist exactly k non-isomorphic groups s such that |g| = |s| and d(g) = d(s). moreover, a 1-fold od-characterizable group is simply called an od-characterizable group. let l = s4(4) be the projective symplectic group in dimension 4 over a field with 4 elements. in this article, we classify groups with the same order and degree pattern as an almost simple group related to l. since aut(l) = z4 hence almost simple groups related to l are l, l : 2 or l : 4. in fact, we prove that l, l : 2 and l : 4 are od-characterizable.