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Easy

Identifying $HK$ and $H\cap K$ for $H$, $K$ Subgroups

ABSALG-XPVLH9

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 $n$ symbols, let $A_n$ be the alternating subgroup of $S_n$, and let $V_4$ be the copy of the Klein $4$-subgroup inside $S_n$ given by $V_4=\{e, (12)(34), (13)(24), (14)(23)\}$.

Which of the following is FALSE?

A

For $G=S_n$, $H=\langle (12)\rangle$, $K=A_n$, we have $HK=G$ and $H\cap K=\{e\}$.

B

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

C

For $G=S_5$, $H=\langle (12345)\rangle$, $K=V_4$, we have $HK=\langle (12345), (2354)\rangle$, and $H\cap K=\{e\}$.

D

For $G=S_4$, $H=\langle (123)\rangle$, $K=V_4$, we have $HK=A_4$ and $H\cap K=\{e\}$.