Finite special linear group SL_2(q)

Let $q=p^f$ , with $p$ prime and $f\geq1$ an integer, and choose the finite field $\mathbb F_q$ . The special linear group $\mathrm{SL}_{2}(q)$ is the group of determinant-one $2\times 2$ matrices over $\mathbb F_q$ , with matrix multiplication.

Order

The group order is $q(q^2-1)$ . The determinant map from the general linear group onto $\mathbb F_q^\times$ has this group as kernel. Dividing the general linear group order by $q-1$ gives the formula.

Simplicity conditions

For odd $q$ , the proper nontrivial center $\{I,-I\}$ prevents simplicity. For even $q$ , the center is trivial and the group equals $\mathrm{PSL}_2(q)$ , which is simple for $q\geq4$ . The remaining case $q=2$ is $S_3$ . Thus simplicity holds exactly when $q$ is even and $q\geq4$ .

Field and category convention

The parameter $q$ must be a prime power. This is a finite abstract group; the field and its vector space supply a construction, rather than a vector-space structure on the matrix group itself.

Determinant sequence

The inclusion into the general linear group and the determinant give an exact sequence

1\longrightarrow\mathrm{SL}_{2}(q)\longrightarrow\mathrm{GL}_{2}(q)\xrightarrow{\det}\mathrm{GL}_1(q)\longrightarrow1.

The diagonal matrix with entries $a,1,\ldots,1$ maps to any prescribed $a\in\mathbb F_q^\times$ , proving surjectivity. In size one, this is the identity map on $\mathbb F_q^\times$ , with trivial kernel.

References

Peter J. Cameron, Notes on Classical Groups, §2.1, Theorems 2.2–2.3, pp. 15–16: orders, determinant and scalar kernels; §2.4, Theorem 2.10, pp. 20–22: simplicity.