# Suzuki groups

In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22n+1), 2B2(22n+1), Suz(22n+1), or G(22n+1), form an infinite family of groups of Lie type found by Suzuki (1960), that are simple for n ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3.

Suzuki (1960) originally constructed the Suzuki groups as subgroups of SL4(F22n+1) generated by certain explicit matrices.

Ree observed that the Suzuki groups were the fixed points of exceptional automorphisms of some symplectic groups of dimension 4, and used this to construct two further families of simple groups, called the Ree groups. In the lowest case the symplectic group B2(2)≈S6; its exceptional automorphism fixes the subgroup Sz(2) or 2B2(2), of order 20. Ono (1962) gave a detailed exposition of Ree's observation.

Tits (1962) constructed the Suzuki groups as the symmetries of a certain ovoid in 3-dimensional projective space over a field of characteristic 2.

Wilson (2010) constructed the Suzuki groups as the subgroup of the symplectic group in 4 dimensions preserving a certain product on pairs of orthogonal vectors.

The Suzuki groups Sz(q) or 2B2(q) are simple for n≥1. The group Sz(2) is solvable and is the Frobenius group of order 20.

The Suzuki groups Sz(q) have orders q2(q2+1)(q−1). These groups have orders divisible by 5, not by 3.

The Schur multiplier is trivial for n>1, Klein 4-group for n=1, i. e. Sz(8).

The outer automorphism group is cyclic of order 2n+1, given by automorphisms of the field of order q.

Suzuki group are Zassenhaus groups acting on sets of size (22n+1)2+1, and have 4-dimensional representations over the field with 22n+1 elements.

Suzuki groups are CN-groups: the centralizer of every non-trivial element is nilpotent.

When n is a positive integer. Sz(q) has at least 4 types of maximal subgroups.

Either q+2r+1 or q-2r+1 is divisible by 5, so that Sz(q) contains the Frobenius group C5:4.

Suzuki (1960) showed that the Suzuki group has q+3 conjugacy classes. Of these q+1 are strongly real, and the other two are classes of elements of order 4.

Suzuki (1960) showed that the Suzuki group has q+3 irreducible representations over the complex numbers, 2 of which are complex and the rest of which are real. They are given as follows: