CHARACTERIZATION OF PROJECTIVE SPECIAL LINEAR GROUPS IN DIMENSION THREE BY THEIR ORDERS AND DEGREE PATTERNS

The prime graph Gamma(G) of a group G is a graph with vertex set pi(G), the set of primes dividing the order of G, and two distinct vertices p and q are adjacent by an edge written p similar to q if there is an element in G of order pq. Let pi(G) = {p(1), p(2), ..., p(k)}. For p is an element of pi(G), set deg(p) := vertical bar{q is an element of pi(G)vertical bar p similar to q}vertical bar, which is called the degree of p. We also set D(G) := (deg(p(1)), deg(p(2)), ..., deg(p(k))), where p(1) < p(2) < .... < p(k), which is called degree pattern of G. The group G is called k-fold OD-characterizable if there exists exactly k non-isomorphic groups M satisfying conditions vertical bar G vertical bar = vertical bar M vertical bar and D(G) = D(M). In particular, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, as the main result, we prove that projective special linear group L-3(2(n)) where n is an element of {4, 5, 6, 7, 8, 10, 12} is OD-characterizable.