Let $G_1$ be a group with neutral element $e_1$ and let $G_2$ be a group with neutral element $e_2$.

Let $f:G_1\rightarrow G_2$ be a group homomorphism.

The **kernel** of $f$, denoted ${\rm Ker}(f)$, is the subset of $G_1$ given by

${\rm Ker}(f):=\{g_1\in G_1\mid f(g_1)=e_2\}$

__Notation__: $S_4$ is the group of permutations on $\{1,2,3,4\}$

Which of the following is **FALSE**?