I've encountered this question: https://math.stackexchange.com/a/267366/352213 and I couldn't see why does $S_4 / V_4 \cong C_6 \Longleftrightarrow A_4 \leq V_4 $ but I can't see why would one imply the other?
2026-03-29 23:45:00.1774827900
Why $S_4 / V_4 \cong C_6 \Longleftrightarrow A_4 \leq V_4 $?
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Since $A_4$ has 12 elements the quotient of $S_4$ by a normal subgroup containing $A_4$ is either $C_2$ or the trivial group. Therefore the two statements are contradictory.