Group theory and generalizations Classification of finite simple groups Object

Suzuki groups

Builds on Group of Lie type Simple group Order of a group Symplectic group

Description

The Suzuki groups $\mathrm{Sz}(q)$ , also written ${}^2B_2(q)$ , are an infinite family of finite groups of Lie type. The parameter is $q = 2^{2n+1}$ with $n \ge 0$ . The order 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; 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 $\mathrm{Sp}_4(\F_q)$ 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 $\mathrm{Suz}$ .

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Tag	`36X3GET13B`
URI	`m2://36X3GET13B`
Kind	object
Writing	W0 stub
MSC	20D06, 20G40
Uses	Group of Lie type, Simple group, Order of a group, Symplectic group
Same as	Wikidata Q7651367 Wikipedia Suzuki groups
Changed	2 November 2026

Suzuki groups

Description

Remarks

Reading path

Further reading

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