Difficult# Group Homomorphism: Kernels and Images for $D_4$ and $Q_8$

ABSALG-ORKYY7

Up to isomorphism, there are two non-abelian groups of order $8$, the dihedral group $D_4$ and the quaternion group $Q_8$, defined in terms of generators and relations by:

$D_4=\langle r,s \mid r^4=s^2=e; \quad srs^{-1}=r^{-1}\rangle,\qquad Q_8=\langle i, j, k\mid i^2=j^2=k^2=ijk=\,-1\rangle$

...where $e$ is the identity (neutral) element of $D_4$ and $1$ is the identity element of $Q_8$, with $(-1)^2=1$.

Let $G$ be a group. By definition, a group homomorphism of $G$ to another group $H$ is a **monomorphism** if and only if it is injective.

It is **trivial** if and only if its image is the identity element of $H$.

Let $\mathcal{K}(G)$ be the set of groups, up to isomorphism, given by the kernels of the **group homomorphisms** of $G$

that are **not monomorphisms** and are **not trivial** homomorphisms .

Let $\mathcal{I}(G)$ be the set of groups, up to isomorphism, given by the images of the **group homomorphisms** of $G$

that are **not monomorphisms** and are **not trivial** homomorphisms .

Which of the following are true?

Select **ALL** that apply.