Assuming we have a homomorphism from $G\rightarrow G$, defined as $f(A) = e_G$. Is that a valid homomorphism? For example, $\mathrm{GL}(2,\mathbb{R}) \rightarrow \mathrm{GL}(2,\mathbb{R})$, defined as $f(A) = I$, where $A$ is a matrix in $\mathrm{GL}(2,\mathbb{R})$.
Am I correct to say that $\ker(f) = G$? Or in my specific example $\ker(f) = \mathrm{GL}(2,\mathbb{R})$? If not, is there a way to construct a homomorphism from a group to itself such that $\ker(f) = G$, or in the specific example I suggested, $\ker(f) = \mathrm{GL}(2,\mathbb{R})$?
Thanks!