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# First Isomorphism Theorem for Groups

ABSALG-4H6XS1

If $f:G_1\rightarrow G_2$ is a group surjection (epimorphism), and $K={\rm Ker}(f)$, then:

the group $G_1/K$ is isomorphic to the group $G_2$

Let $Q$ be the quaternion group given by

$Q=\{\pm 1, \pm I,\pm J,\pm K\}$

...with composition rules:

$I^2=J^2=K^2=-1, IJ=K=-JI, JK=I=-KJ, KI=J=-IK$

Let $D_3$ be the dihedral group given by

$D_3=\{1,r,r^2,s,sr,sr^2\}$

...with composition rules:

$r^3=s^2=(sr)^2=1$

Let $f:Q\rightarrow G_2$ be any group surjection with kernel $\{\pm 1\}$.

Denote by $S_n$ the symmetric group on $n\ge 1$ letters.

Which of the following is false?

A

The group $G_2$ is isomorphic to the subgroup ${\mathcal V}_4=\{1,(12),(34),(12)(34)\}$ of $S_4$

B

The group $G_2$ is abelian

C

Every non-neutral element of $G_2$ has order $2$

D

There is no group homomorphism $g:S_4\rightarrow D_3$ with kernel isomorphic to $G_2$

E

There is a group homomorphism $g:S_4\rightarrow S_3$ with kernel isomorphic to $G_2$