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# Isomorphism Theorems (Groups):$HK/K$, for $H, K\le G,K$ Normal

ABSALG-SUUYTI

Let $G$ be a finite group with neutral element $e$ and let $H, K$ be subgroups of $G$.

For $g\in G$, let $\langle g\rangle$ be the subgroup of $G$ generated by $g$.

Let $S_n$, $n\ge 4$, be the symmetric group on the set $\{1, 2, \ldots, n\}$, and let $A_n$ be the alternating subgroup.

Let $A_4$ be the alternating subgroup on $\{1, 2, 3, 4\}$ and $V_4=\{e, (12)(34), (13)(24), (14)(23)\}$, both viewed as subgroups of $S_n$.

Which of the following is false?

A

For $G=S_4$, $H=\{e, (12), (34), (12)(34)\}$, $K=V_4$, we have $HK/K\simeq\mathbb{Z}/2\mathbb{Z}$.

B

For $G=S_4$, $H=\{e, (12), (34), (12)(34)\}$, $K=A_4$, we have $HK/K\simeq\mathbb{Z}/2\mathbb{Z}$.

C

For $G=S_5$, $H=\langle (12)\rangle$, $K=A_4$, we have $HK/K\simeq\mathbb{Z}/2\mathbb{Z}$.

D

For $G=S_5$, $H=\langle (15)\rangle$, $K=A_4$, we have $HK/K\simeq\mathbb{Z}/2\mathbb{Z}$.

E

For $G=S_5$, $H=\langle (123)\rangle$, $K=V_4$, we have $HK/K\simeq\mathbb{Z}/3\mathbb{Z}$.