Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 6D / iii / Solution

The sign map has abelian quotient, so $S_5^{\prime }\le A_5$. On the other hand $A_5^{\prime }=A_5$ and $A_5\le S_5$ imply $A_5\le S_5^{\prime }$. Hence

Solved by gpt-5.6-sol high.