Exercise 8.4.2

Proof.

We notice that for all $i,j,k$ such that $i\ne j,j\ne k$, $(ij)(jk)=(\mathit{ijk})$.

As every permutation in $A_n$ is the product of an even number of permutations, it is sufficient to prove that the product of $(ij)(kl),i\ne j,k\ne l$, is a product of 3-cycles.

$\text{∙}$ If $\{i,j\},\{k,l\}$ are disjointed, then $i,j,k,l$ are distincts, so

$\text{∙}$ If $\{i,j\},\{k,l\}$ have one common element, say $i=k,i\ne l$, then

is a 3-cycle.

$\text{∙}$ If $\{i,j\}=\{k,l\}$, then $(ij)(kl)={(ij)}^2=()=e$ is the empty product.

Conclusion: $A_n$ is generated by 3-cycles. □