# Suzuki groups

In the area of modern algebra known as group theory, the Suzuki groups, denoted by Suz(22n+1), Sz(22n+1), G(22n+1), or 2B2(22n+1), form an infinite family of groups of Lie type found by Suzuki (1960), that are simple for n ≥ 1.

## Constructions

### Suzuki

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

### Ree

Ree observed that the Suzuki groups were the fixed points of an exceptional automorphism of the symplectic groups in 4 dimensions, and used this to construct two further families of simple groups, called the Ree groups. Ono (1962) gave a detailed exposition of Ree's observation.

### Tits

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

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.

## Properties

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

The Suzuki groups have orders q2(q2+1) (q−1) where q = 22n+1. They are the only non-cyclic finite simple groups of orders not divisible by 3.

The Schur multiplier is trivial for n≠1, elementary abelian of order 4 for 2B2(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.

### Conjugacy classes

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.

The non-trivial elements of the Suzuki group are partitioned into the non-trivial elements of nilpotent subgroups as follows (with r=2n, q=22n+1):

• q2+1 Sylow 2-subgroups of order q2, of index q–1 in their normalizers. 1 class of elements of order 2, 2 classes of elements of order 4.
• q2(q2+1)/2 cyclic subgroups of order q–1, of index 2 in their normalizers. These account for (q–2)/2 conjugacy classes of non-trivial elements.
• Cyclic subgroups of order q+2r+1, of index 4 in their normalizers. These account for (q+2r)/4 conjugacy classes of non-trivial elements.
• Cyclic subgroups of order q–2r+1, of index 4 in their normalizers. These account for (q–2r)/4 conjugacy classes of non-trivial elements.

The normalizers of all these subgroups are Frobenius groups.

### Characters

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:

• The trivial character of degree 1.
• The Steinberg representation of degree q2, coming from the doubly transitive permutation representation.
• (q–2)/2 characters of degree q2+1
• Two complex characters of degree r(q–1) where r=2n
• (q+2r)/4 characters of degree (q–2r+1)(q–1)
• (q–2r)/4 characters of degree (q+2r+1)(q–1).