In this short paper we have characterized Suzuki's simple groups SZ(22m+l), m > 1 using only the set 7re(G) of orders of elements in the group G. That is, we have Theorem 2. Let G be a finite group. Then G S (22m+l), m > 1 if and only if ir,(G) = {2, 4, all factors of (22m+l 1), (22m+l 2m+l + 1), and (22m+l + 2m+1 + 1)}. Suzuki's simple groups Sz(22m+l), m > 1 is a family of Zassenhaus groups (Z-groups) of odd degree [9]. In [4] we characterized another family of Zassenhaus groups of odd degree L2(2m) using only the set of orders of elements in the group G. That is, let 7re(G) denote the set of orders of elements in the group G. Then we have proved the following theorem. Theorem 1. Let G be a finite group. Then G L2(2m), m > 2 if and only if 7te(G) = {2, all factors of (2m 1) and 2m + 1)}. In this short paper, we continue this work and obtain the following theorem. Theorem 2. Let G be a finite group. Then G S_(22m+l), m > 1 if and only if 7re(G) = {2, 4, all factors of (22m+1 1), (22m+1 2m+1 + 1), and (22m+1 + 2m+1 + 1)}. Since the simple Z-groups of odd degree consists of L2(2m)(m > 2) and Sz(22m+l), (m > 1), we have Corollary. Let G be a finite group and M a simple Z-groups of odd degree. Then G M if and only if re (G) = 7te(M). Before starting the proof we give a remark about the set 7re(G) in Theorem 2. Since 22m+ I1 4 0 (mod 3) and (22m+I 2m+I + 1) * (22m+I + 2m+I + 1) = 24m+2+ 1 $ 0 (mod 3), 3 ? 7re(G). And 22m+I -1 is prime to 5, but 5 E ire(G) by 24m+2 + 1 = 0 (mod 5). Proof of Theorem 2. We need only prove the sufficiency by [3, XI Theorem 3.10]. Received by the editors June 25, 1990 and, in revised form, October 9, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 20D05, 20D06. I The author would like to thank the Department of Pure Mathematics of the University of Sydney for the hospitality. ? 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page