# On the Suzuki group

Let $G$ be the Suzuki group over the field with $q=2^{2m+1}$ elements, $m>0$. Then, by Theorem 3.10 from B. Huppert, N. Blackburn, Finite Group III, pp 192-193, or wikipedia, the group $G$ contains subgroups $F,A,B,$ and $C$ such that

$$\{A^x\setminus \{1\}, B^x\setminus \{1\}, C^x\setminus \{1\}, F^x\setminus \{1\},\mid x \in G\}$$

is a partition for $G\setminus \{1\}$. Here $F$ is a sylow 2-subgroup of $G$ of order $q^2$ and $|\{F^x\setminus \{1\}\mid x \in G\}|=q^2+1$. Also $A$ is a cyclic group of order $q-1$ and $|\{A^x\setminus \{1\}\mid x \in G\}|=q^2(q^2+1)/2$. The value of $|\{B^x\setminus \{1\}\mid x \in G\}|=r$ and $|\{C^x\setminus \{1\}\mid x \in G\}|=s$ are not mention. What is the value of $r$ and $s$? Thank for any helps.

• John Britnell, Azizollah Azad and I made use of this partition in this paper - arxiv.org/pdf/1305.3748.pdf - if you read Section 5.5 you should find the information you need. – Nick Gill Jun 30 '14 at 15:22