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. These simple groups are the only finite non-abelian ones with orders not divisible by 3.

Constructions

Suzuki

Suzuki (1960) originally constructed the Suzuki groups as subgroups of SL₄(F_2²ⁿ⁺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. These groups have orders divisible by 5, not 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.

Subgroups

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).

References

Nouacer, Ziani (1982), "Caractères et sous-groupes des groupes de Suzuki", Diagrammes, 8: ZN1--ZN29, ISSN 0224-3911, MR 0780446
Ono, Takashi (1962), "An identification of Suzuki groups with groups of generalized Lie type.", Annals of Mathematics. Second Series, 75: 251–259, doi:10.2307/1970173, ISSN 0003-486X, JSTOR 1970173, MR 0132780
Suzuki, Michio (1960), "A new type of simple groups of finite order", Proceedings of the National Academy of Sciences of the United States of America, 46: 868–870, doi:10.1073/pnas.46.6.868, ISSN 0027-8424, JSTOR 70960, MR 0120283, PMC 222949, PMID 16590684
Suzuki, Michio (1962), "On a class of doubly transitive groups", Annals of Mathematics. Second Series, 75: 105–145, doi:10.2307/1970423, ISSN 0003-486X, JSTOR 1970423, MR 0136646
Tits, Jacques (1962), "Ovoïdes et groupes de Suzuki", Archiv der Mathematik, 13: 187–198, doi:10.1007/BF01650065, ISSN 0003-9268, MR 0140572
Wilson, Robert A. (2010), "A new approach to the Suzuki groups", Mathematical Proceedings of the Cambridge Philosophical Society, 148 (3): 425–428, doi:10.1017/S0305004109990399, ISSN 0305-0041, MR 2609300