---
tag: 36X3GET13B
kind: object
name: Suzuki groups
msc: [20D06, 20G40]
uses: [NPEBABQ49M, 0YMYMNQ0QZ, 0B85S2V2DR, X12FS2JAP0]
sources:
  - url: https://en.wikipedia.org/wiki/Suzuki_groups
same_as:
  wikidata: Q7651367
  wikipedia: Suzuki groups
---

## Description

The Suzuki groups $\mathrm{Sz}(q)$, also written ${}^2B_2(q)$, are an infinite family of finite [[groups of Lie type|NPEBABQ49M]].
The parameter is $q = 2^{2n+1}$ with $n \ge 0$.
The [[order|0B85S2V2DR]] of $\mathrm{Sz}(q)$ is
$$\abs{\mathrm{Sz}(q)} = q^2 (q^2 + 1)(q - 1).$$
For $n \ge 1$ the group $\mathrm{Sz}(q)$ is a non-abelian [[simple group|0YMYMNQ0QZ]]; the smallest one is $\mathrm{Sz}(8)$, of order $29120$.
The group $\mathrm{Sz}(2)$ has order $20$ and is not simple.

## Remarks

$\mathrm{Sz}(q)$ is the subgroup of the [[symplectic group|X12FS2JAP0]] $\mathrm{Sp}_4(\F_q)$ fixed by a certain automorphism.
This automorphism combines the exceptional graph automorphism of $\mathrm{Sp}_4$ in characteristic $2$ with a field automorphism, and it exists because $q$ is an odd power of $2$.
It acts doubly transitively on a set of $q^2 + 1$ points, an ovoid in the projective space of dimension $3$ over $\F_q$.
The Suzuki groups are the only non-abelian finite simple groups whose order is not divisible by $3$.
They were found by Michio Suzuki.
They are not related to the sporadic Suzuki group $\mathrm{Suz}$.

## Explanations

### L1 human

Most finite simple groups have an order divisible by $3$.
The Suzuki groups are the exception: an infinite family of simple groups whose sizes are not multiples of $3$.
They live inside $4 \times 4$ symplectic matrix groups over fields with $2^{2n+1}$ elements, and they exist because of a special symmetry that appears only in characteristic $2$.