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The Möbius group is the group of all Möbius transformations of the Riemann sphere, under function composition. An invertible $2\times 2$ complex matrix represents $z\mapsto (az+b)/(cz+d)$, and nonzero scalar multiples represent the same transformation. Rescaling a representative to determinant one gives a surjective group homomorphism $S{L}_2(\mathbb{C})\to \mathcal{M}$ with kernel of a group homomorphism $\{I,-I\}$. Thus it is also the projective linear group $PG{L}_2(\mathbb{C})$ and the quotient $PS{L}_2(\mathbb{C})=S{L}_2(\mathbb{C})/\{\pm I\}$.

A scalar matrix is a scalar multiple of the identity matrix, equivalently a diagonal matrix with all diagonal entries equal. It commutes with every square matrix of the same size. In $G{L}_n(F)$, the nonzero scalar matrices form the central subgroup removed to define the projective linear group.

For any field $F$, a projective linear group element is uniquely determined by its images of three distinct points of the projective line, and any ordered distinct target triple is attainable. Given representatives $v_z,v_x$ forming a basis and $v_y=A{v}_z+B{v}_x$, the matrix with columns $A{v}_z,B{v}_x$ sends $(0,1,\infty )$ to $(x,y,z)$. Distinctness forces $A,B\ne 0$. The matrices realizing one projective map differ by a nonzero scalar; over ${\mathbb{F}}_p$ there are $p-1$ such matrices.