Kernel (algebra) in the context of Injective function

In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that

where the group operation on the left side of the equation is that of G and on the right side that of H.

In mathematics, the special linear group ${\displaystyle \operatorname {SL} (n,R)}$ of degree ${\displaystyle n}$ over a commutative ring ${\displaystyle R}$ is the set of ${\displaystyle n\times n}$ matrices with determinant ${\displaystyle 1}$, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant

where ${\displaystyle R^{\times }}$ is the multiplicative group of ${\displaystyle R}$ (that is, ${\displaystyle R}$ excluding ${\displaystyle 0}$ when ${\displaystyle R}$ is a field).