ODs-CHARACTERIZATION OF SOME LOW-DIMENSIONAL FINITE CLASSICAL GROUPS

The solvable graph of a finite group G, which is denoted by Γs(G), is a simple graph whose vertex set is comprised of the prime divisors of |G| and two distinct primes p and q are joined by an edge if and only if there exists a solvable subgroup of G such that its order is divisible by pq. Let p1 < p2 < · · · < pk be all prime divisors of |G| and let Ds(G) = (ds(p1), ds(p2), . . . , ds(pk)), where ds(p) signifies the degree of the vertex p in Γs(G). We will simply call Ds(G) the degree pattern of solvable graph of G. A finite group H is said to be ODs-characterizable if H ∼= G for every finite group G such that |G| = |H| and Ds(G) = Ds(H). In this paper, we study the solvable graph of some subgroups and some extensions of a finite group. Furthermore, we prove that the linear groups L3(q) with certain properties, are ODs-characterizable.