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Abstract Algebra

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Is It Semidirect?

ABSALG-FF1EA4

In which of the following cases is $G$ a semidirect product of $H$ and $K$?

Note that $V_4=\{(1), (12)(34), (13)(24), (14)(23)\}$ is the Klein group.

A

$G=A_4$, $H=\langle (123)\rangle$, $K=V_4$

B

$G=Q_8$, $H=\langle i\rangle$, $K=\langle -1\rangle$

C

$G=\mathbb{Z}_2\times\mathbb{Z}_4$, $H=\langle (1,2)\rangle$, $K=\langle (0,2)\rangle$

D

$G=A_5$, $H=\langle (12)(34)\rangle$, $K=\langle (234)\rangle$