Finite matrix group GL_2(3)

The finite matrix group $\mathrm{GL}_2(3)$ consists of invertible $2\times2$ matrices over the field $\mathbb F_3$ , with matrix multiplication.

Order and nonsimplicity

The independent-column count gives $(3^2-1)(3^2-3)=48$ elements. The determinant maps onto $\mathbb F_3^\times\cong C_2$ , with kernel $\mathrm{SL}_2(3)$ of order 24. This proper nontrivial normal subgroup proves nonsimplicity.

Determinant quotient

Identifying $\mathbb F_3^\times=\{1,-1\}$ with the additive group $C_2$ , the determinant is a surjective homomorphism $\mathrm{GL}_2(3)\to C_2$ . Its kernel is $\mathrm{SL}_2(3)$ .

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.