Group homomorphisms

1. $f(e)=f(ee)=f(e)f(e)$, so multiplying by $f(e)^{-1}$: $e'=f(e)$.

2. $f(g)f(g^{-1})=f(gg^{-1})=f(e)=e'$, so $f(g^{-1})=f(g)^{-1}$.

3. $\ker f=\{g:f(g)=e'\}$. Contains $e$ (by 1). If $a,b\in\ker f$: $f(ab^{-1})=f(a)f(b)^{-1}=e'(e')^{-1}=e'$, so $ab^{-1}\in\ker f$. Subgroup.

4. $\operatorname{Im} f=\{f(g):g\in G\}$. Nonempty (contains $e'$). If $f(a),f(b)\in\operatorname{Im}$, then $f(a)f(b)^{-1}=f(ab^{-1})\in\operatorname{Im}$. Subgroup.

Conclusion. All four properties hold.