A characterization of Suzuki’s simple groups

Abstract

In this short paper we have characterized Suzuki’s simple groups S z ( 2 2 m + 1 ) , m ⩾ 1 {S_z}({2^{2m + 1}}),m \geqslant 1 using only the set π e ( G ) {\pi _e}(G) of orders of elements in the group G G . That is, we have Theorem 2. Let G G be a finite group. Then G ≃ S z ( 2 2 m + 1 ) , m ⩾ 1 G \simeq {S_z}({2^{2m + 1}}),m \geqslant 1 if and only if π e ( G ) = { 2 , 4 , a l l f a c t o r s o f ( 2 2 m + 1  - 1),( 2 2 m + 1 − 2 m + 1 + 1 ) , a n d ( 2 2 m + 1 + 2 m + 1 + 1 ) } {\pi _e}(G) = \{ 2,4,all\;factors\;of\;{\text {(}}{{\text {2}}^{2m + 1}}{\text { - 1),(}}{{\text {2}}^{2m + 1}} - {2^{m + 1}} + 1),\;and\;({2^{2m + 1}} + {2^{m + 1}} + 1)\} .