Suzuki groups

This article is about the infinite family of groups of Lie type found by Suzuki. For the sporadic simple group, see Suzuki sporadic group.

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.

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 780446
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, 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, MR 0120283
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, 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

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