Suzuki group

A simple finite group , a member of the infinite series of simple groups, discovered by M. Suzuki.

Let be a natural number, let be the finite field with elements, and let be an automorphism of such that for any . The Suzuki group is then generated by the subgroup consisting of all diagonal matrices of order 4 with diagonal elements (, ), the subgroup consisting of all triangular matrices of the form

, and the matrix

The subgroup is a Sylow -subgroup of the group ; it is a Suzuki -group. The subgroup coincides with the normalizer of the subgroup . The permutation representation of the group on the cosets of is doubly transitive; its degree is equal to . The order of the Suzuki group is equal to , and is not divisible by 3. On the other hand, any non-Abelian finite simple group whose order is not divisible by 3 is isomorphic to a Suzuki group. The group is a maximal subgroup of the symplectic group and is the centralizer in of an automorphism of order 2 of the group . In other words, is isomorphic to — the twisted analogue of a Chevalley group of type over the field with elements.

References

[1]	M. Suzuki, "On a class of doubly transitive groups" Ann. of Math. , 75 : 1 (1962) pp. 105–145
[2]	R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972)

Comments

There is in fact precisely one automorphism of such that for all . It is .

There is a twisted polarity whose absolute points are the -dimensional subspaces of that are fixed by a conjugate of . The set of absolute points thus obtained is an ovoid. See [a1]–[a2].

References

[a1]	J. Tits, "Ovoïdes et groupes de Suzuki" Arch. Math. , 13 (1962) pp. 187–198
[a2]	J. Tits, "Une propriété charactéristique des ovoïdes associés aux groupes de Suzuki" Arch. Math. , 17 (1966) pp. 136–153
[a3]	B. Huppert, "Finite groups" , 3 , Springer (1982) pp. Chapt. IX.3