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# Suzuki groups

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.

Contents

Constructions

Suzuki

Suzuki (1960) originally constructed the Suzuki groups as subgroups of SL4(F22n+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 ^{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}. 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 ^{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.

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*=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.

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
*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).

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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