Constructions
Suzuki
originally constructed the Suzuki groups as subgroups of SL4(F22''n''+1) generated by certain explicit matrices.Ree
Ree observed that the Suzuki groups were the fixed points of exceptional automorphisms of someTits
constructed the Suzuki groups as the symmetries of a certain ovoid in 3-dimensional projective space over a field of characteristic 2.Properties
Let ''q'' = 22''n''+1 and ''r'' = 2''n'', where ''n'' is a non-negative integer. The Suzuki groups Sz(''q'') or 2''B''2(''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 ''q''2(''q''2+1)(''q''−1). These groups have orders divisible by 5, but not by 3. TheSubgroups
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 ''q''2·(''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''+2''r''+1:4 * ''C''''q''–2''r''+1:4 * Smaller Suzuki groups, when 2''n''+1 is composite. Either ''q''+2''r''+1 or ''q''–2''r''+1 is divisible by 5, so that Sz(''q'') contains the Frobenius group ''C''5:4.Conjugacy classes
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. *''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.Characters
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
* * * * * *{{Citation , last1=Wilson , first1=Robert A. , title=A new approach to the Suzuki groups , doi=10.1017/S0305004109990399 , mr=2609300 , year=2010 , journal=Mathematical Proceedings of the Cambridge Philosophical Society , issn=0305-0041 , volume=148 , issue=3 , pages=425–428, s2cid=18046565External links
* http://brauer.maths.qmul.ac.uk/Atlas/v3/exc/Sz8/ * http://brauer.maths.qmul.ac.uk/Atlas/v3/exc/Sz32/ Finite groups