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Abstract Algebra

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Group Homomorphism: Kernel and Image of a Homomorphism of $S_5$

ABSALG-FGV9XN

Let $S_5$ be the symmetric group on $5$ symbols, represented as the group of permutations of the set $\{1,2,3,4,5\}$.
Let $(\cdot)$ be the neutral element of $S_5$ and let $A_5$ be the alternating subgroup of $S_5$.

Let $\varphi:S_5\rightarrow G$ be a group homomorphism of $S_5$ 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 $\varphi$.