Easy# Group Homomorphism: Kernel and Image of a Homomorphism of $S_3$

ABSALG-LXDRSB

Let $S_3$ be the symmetric group on $3$ symbols, represented as the group of permutations of the set $\{1,2,3\}$.

Let $(\cdot)$ be the neutral element of $S_3$.

Let $\varphi:S_3\rightarrow G$ be a group homomorphism of $S_3$ to another group $G$.

The group

**can not be** isomorphic to the kernel of $\varphi$.

The group

**can not be** isomorphic to the image of $\varphi$.