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# 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 $\varphi$.