Show that $(123)$ and $(23\ldots n)$ together generate $A_n$ if $n$ is even. I know what to do for $(123)$ and $(12\ldots n)$ with n is odd, but how do I do this one?
2026-04-23 09:28:49.1776936529
Show that $(123) and (23\ldots n)$ together generate $A_n$ if n is even.
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Hint: $A_{n-1}$ is a subgroup of $A_n$ (consider permutations in $A_n$ that leave $1$ fixed).