Suzuki group

A simple finite group $ \mathop{\rm Sz} ( q) $, a member of the infinite series of simple groups, discovered by M. Suzuki.

Let $ n $ be a natural number, let $ F $ be the finite field with $ q = 2 ^ {2n+} 1 $ elements, and let $ \theta $ be an automorphism of $ F $ such that $ \alpha ^ {\theta ^ {2} } = \alpha ^ {2} $ for any $ \alpha \in F $. The Suzuki group $ \mathop{\rm Sz} ( q) $ is then generated by the subgroup $ T $ consisting of all diagonal matrices of order 4 with diagonal elements $ \lambda ^ {1+ \theta } , \lambda , \lambda ^ {-} 1 , ( \lambda ^ {1+ \theta } ) ^ {-} 1 $( $ \lambda \in F $, $ \lambda \neq 0 $), the subgroup $ U $ consisting of all triangular matrices of the form

$$ \left \| \begin{array}{cccc} 1 & 0 & 0 & 0 \\ \alpha & 1 & 0 & 0 \\ \alpha ^ {1+ \theta } + \beta &\alpha ^ \theta & 1 & 0 \\ \alpha ^ {2+ \theta } + \alpha \beta + \beta ^ \theta &\beta &\alpha & 1 \\ \end{array} \right \| $$

$ ( \alpha , \beta \in F ) $, and the matrix

$$ \left \| \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right \| . $$

The subgroup $ U $ is a Sylow $ 2 $- subgroup of the group $ \mathop{\rm Sz} ( q) $; it is a Suzuki $ 2 $- group. The subgroup $ UT $ coincides with the normalizer of the subgroup $ U $. The permutation representation of the group $ \mathop{\rm Sz} ( q) $ on the cosets of $ UT $ is doubly transitive; its degree is equal to $ q ^ {2} + 1 $. The order of the Suzuki group $ \mathop{\rm Sz} ( q) $ is equal to $ q ^ {2} ( q- 1)( q ^ {2} + 1) $, and is not divisible by 3. On the other hand, any non-Abelian finite simple group whose order is not divisible by 3 is isomorphic to a Suzuki group. The group $ \mathop{\rm Sz} ( q) $ is a maximal subgroup of the symplectic group $ \mathop{\rm Sp} ( 4, q) $ and is the centralizer in $ \mathop{\rm Sp} ( 4, q) $ of an automorphism of order 2 of the group $ \mathop{\rm Sp} ( 4, q) = B _ {2} ( q) $. In other words, $ \mathop{\rm Sz} ( q) $ is isomorphic to $ {} ^ {2} B _ {2} ( q) $— the twisted analogue of a Chevalley group of type $ B _ {2} $ over the field with $ q $ elements.

References

[1]	M. Suzuki, "On a class of doubly transitive groups" Ann. of Math. , 75 : 1 (1962) pp. 105–145
[2]	R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972)

Comments

There is in fact precisely one automorphism $ \theta $ of $ F $ such that $ \theta ^ {2} ( \alpha ) = \alpha ^ {2} $ for all $ \alpha \in F $. It is $ \theta ( \alpha ) = \alpha ^ {2 ^ {m+ 1 } } $.

There is a twisted polarity whose absolute points are the $ q ^ {2} + 1 $ $ 1 $- dimensional subspaces of $ F _ {q} ^ {q} $ that are fixed by a conjugate of $ UT $. The set of absolute points thus obtained is an ovoid. See [a1]–[a2].

References

[a1]	J. Tits, "Ovoïdes et groupes de Suzuki" Arch. Math. , 13 (1962) pp. 187–198
[a2]	J. Tits, "Une propriété charactéristique des ovoïdes associés aux groupes de Suzuki" Arch. Math. , 17 (1966) pp. 136–153
[a3]	B. Huppert, "Finite groups" , 3 , Springer (1982) pp. Chapt. IX.3