Suzuki groups

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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.

Constructions[edit]

Suzuki[edit]

Suzuki (1960) originally constructed the Suzuki groups as subgroups of SL4(F22n+1) generated by certain explicit matrices.

Ree[edit]

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[edit]

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[edit]

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[edit]

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. They are the only non-cyclic finite simple groups of orders not divisible 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[edit]

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[edit]

Characters[edit]

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[edit]