Abstract Algebra

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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.

A

$\mathcal{K}(D_4)\not=\mathcal{K}(Q_8)$

B

$\mathcal{I}(D_4)\not=\mathcal{I}(Q_8)$

C

$\mathcal{K}(D_4)=\mathcal{K}(Q_8)$

D

$\mathcal{I}(D_4)=\mathcal{I}(Q_8)$