I need to show that:
- $S=\left\{(12),(13),...,(1n)\right\}$ generates $S_n$
- $S=\left\{(12),(123\cdots n)\right\}$ generates $S_n$
How do I show that each one of them generates $S_n$?
Thank you!
2026-04-03 18:01:01.1775239261
How to show that two groups makes $S_n$
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Can you prove that every element of $S_n$ is equal to a product of transpositions? If so, you just need to show that each of those generating sets contains all transpositions.
Edit: To make this more direct, this task becomes straight forward once you understand how conjugation works in $S_n$.