Suzuki groups

In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(2²ⁿ⁺¹), ²B₂(2²ⁿ⁺¹), Suz(2²ⁿ⁺¹), or G(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 exceptional automorphisms of some symplectic groups of dimension 4, and used this to construct two further families of simple groups, called the Ree groups. In the lowest case the symplectic group B₂(2)≈S₆; its exceptional automorphism fixes the subgroup Sz(2) or ²B₂(2), of order 20. 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

Let q = 2²ⁿ⁺¹ and r = 2ⁿ, where n is a non-negative integer.

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

The Suzuki groups Sz(q) have orders q²(q²+1)(q−1). These groups have orders divisible by 5, but not by 3.

The Schur multiplier is trivial for n>1, Klein 4-group for n=1, i. e. Sz(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.

Subgroups

When n is a positive integer, Sz(q) has at least 4 types of maximal subgroups.

The diagonal subgroup is cyclic, of order q – 1.

The lower triangular (Borel) subgroup and its conjugates, of order q²·(q-1). They are one-point stabilizers in a doubly transitive permutation representation of Sz(q).
The dihedral group D_q–1, normalizer of the diagonal subgroup, and conjugates.
C_q+2r+1:4
C_q–2r+1:4
Smaller Suzuki groups, when 2n+1 is composite.

Either q+2r+1 or q–2r+1 is divisible by 5, so that Sz(q) contains the Frobenius group C₅:4.

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.

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).

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 (2): 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 (6): 868–870, Bibcode:1960PNAS...46..868S, 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 (1): 105–145, doi:10.2307/1970423, hdl:2027/mdp.39015095249804, 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, S2CID 121482873
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, S2CID 18046565