# 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)

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].