Sz(2) - Knowlpedia

For $q=2=2^{2m+1}$ , where $m=0$ , ${}^2B_2(2)$ is the Steinberg fixed-point group

{}^2B_2(2)=\mathbf G_{B_2,\mathrm{sc}}^F,\qquad F=\tau F_{2^m},\qquad \tau^2=F_2.

Here $\mathbf G_{B_2,\mathrm{sc}}$ is the split simply connected algebraic group of type $B_2$ over $\overline{\mathbb F}_2$ , and $\tau$ is its exceptional endomorphism interchanging the long and short root data. It commutes with Frobenius. Thus $F^2=F_q$ ; the group consists of all $g$ with $F(g)=g$ , under inherited multiplication. Its center is trivial.

Order and simplicity

Its order is

|{}^2B_2(2)|=20.

It is not simple: Sz(2) is a solvable Frobenius group of order 20. The exact order is 20.

Parameter convention

The absolute root-system rank is $2$ , and the exponent of $2$ in $q$ must be odd. The notation does not define this family over arbitrary prime powers. The traditional notation $\operatorname{Sz}(q)$ is another name for ${}^2B_2(q)$ .

The prime-field boundary

The same fixed-point construction at $m=0$ gives $\operatorname{Sz}(2)$ , of order 20, isomorphic to $C_5\rtimes C_4$ with faithful action of $C_4=\operatorname{Aut}(C_5)$ . It is excluded from the simple-family tile.

References

Robert A. Wilson, On the simple groups of Suzuki and Ree (24 April 2010), Introduction; §5.1 orders, §5.2 simplicity, and §5.3 the groups over the prime fields, pp. 24–30.
Robert Steinberg, Lectures on Chevalley Groups (Yale, 1967), §11, especially Theorems 34–35, PDF pp. 193–196; notation for field and twisting parameters differs.
B. Akbari, ODs-characterization of some low-dimensional finite classical groups, Table 2, p. 85: order column only; its additional Restrictions concern a separate invariant.