In the area of modern algebra known as group theory, the **Suzuki groups**, denoted by Suz(2^{2n+1}), Sz(2^{2n+1}), *G*(2^{2n+1}), or ^{2}*B*_{2}(2^{2n+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 SL_{4}(**F**_{22n+1}) generated by certain explicit matrices.

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 (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 are simple for *n*≥1. The group ^{2}*B*_{2}(2) is solvable and is the Frobenius group of order 20.

The Suzuki groups have orders *q*^{2}(*q*^{2}+1) (*q*−1) where *q* = 2^{2n+1}. These groups have orders divisible by 5, not by 3.

The Schur multiplier is trivial for *n*≠1, elementary abelian of order 4 for ^{2}*B*_{2}(8).

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

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

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

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*=2^{n}, *q*=2^{2n+1}):

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

The normalizers of all these subgroups are Frobenius groups.

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 *q*^{2}, coming from the doubly transitive permutation representation.
(*q*–2)/2 characters of degree *q*^{2}+1
Two complex characters of degree *r*(*q*–1) where *r*=2^{n}
(*q*+2*r*)/4 characters of degree (*q*–2*r*+1)(*q*–1)
(*q*–2*r*)/4 characters of degree (*q*+2*r*+1)(*q*–1).