On the Sylow Normalizers of Some Simple Classical Groups

Let G be a finite group and π(G) be the set of prime divisors of the order of G. For t ϵ π(G) denote by n_t (G) the order of a normalizer of t-Sylow subgroup of G and put n(G) {nt (G) : t є π(G)g. In this paper, we give an answer to the following problem, for the groups of Lie type Bn, Cn and Dn: “Let L be a finite non-abelian simple group and G be a finite group with n(L) = n(G). Is it true that L ≡ G?” In this paper, we find the first examples of non-abelian finite simple groups which are not isomorphic and they have the same set of orders of Sylow normalizers and hence, we show that the question above is not correct always. Let ₰ be the set of prime numbers of order 2n, 2(n-1) and 2(n-2) mod q. The latter condition is necessary if n ≥ 5. Also, we show that Dn+1(q) is determined uniquely by its order and fnt (Dn+1(q)) : t є ₰ u{2} and if n = 2 or q ≠+-1 (mod 8), then Bn(q) and Cn(q) are characterizable by their orders and orders of t-Sylow normalizers, where t є ₰ u{2}. If n≥ 3 and q≠+-1 (mod 8), then Bn(q) andCn(q) are 2-characterizable by their orders and the orders of t-Sylow normalizers, where t є ₰ u{2}.