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Identifying Groups of Order $16$ using Kernels

ABSALG-BHB8YT

Let $G$ be a group of order $16$, with neutral element $e$.

Suppose that $G$ is generated by two elements $r, s\in G$ subject to certain relations.

Suppose, in addition, that there are three surjective homomorphisms $\varphi_i:G\rightarrow \mathbb{Z}/2\mathbb{Z}$, $i=1,2,3$, with kernels:

$$\ker(\varphi_1)=\langle r\rangle, \quad \ker(\varphi_2)=\langle r^2, s\rangle,\quad \ker(\varphi_3)=\langle r^2, rs\rangle$$

...no two of which are isomorphic subgroups of $G$.

Which of the following groups satisfies all the above properties?

A

$D_8:=\langle r^8=s^2=e,\;{srs}^{-1}=r^{-1}\rangle$

B

$SD_8:=\langle r^8=s^2=e,\;srs^{-1}=r^3\rangle$

C

$Q_{16}=\langle r^4=s^2,\;r^8=e,\;sr=r^{-1}s\rangle$

D

$\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$