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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
Select Option $\{(\cdot)\}$$\small\{(\cdot), (12)(34), (13)(24), (14)(23)\}\normalsize$$A_5$$S_5$
can not be isomorphic to the kernel of $\varphi$. The group
Select Option $\{(\cdot)\}$$\mathbb{Z}/2\mathbb{Z}$$\mathbb{Z}/5\mathbb{Z}\times S_3$$S_5$
can not be isomorphic to the image of $\varphi$.
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