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# Quotient of Group by Normal Subgroup: Group Homomorphisms

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Let $G$ be a group with neutral element $e$, and let $K$ be a normal subgroup of $G$, written $K\trianglelefteq G$.

The quotient group $G/K$ is the set of right cosets (=left cosets, as $K$ is normal) of $K$ in $G$,

with composition $KgKh=Kgh$ for $g,h\in G$.

In the last two statements of the ​following question $S_4$ denotes the symmetric group on $4$ symbols.

True

False

True

False

Every group $G$ is isomorphic to a quotient of itself.

True

False

$K\trianglelefteq G$ if and only if $K=\ker(\varphi)$ for some group surjection $\varphi$.

True

False

The group $\mathbb{Z}/2\mathbb{Z}$ is isomorphic to a normal subgroup of $S_4$

True

False

The group $\mathbb{Z}/2\mathbb{Z}$ is not isomorphic to any quotient of $S_4$.