# Thread: group permutations 6

1. ## group permutations 6

If n >= 3, show that every element in An is a product of 3-cycles.

Please show steps. Thanks!

2. Originally Posted by mpryal
If n >= 3, show that every element in An is a product of 3-cycles.

Please show steps. Thanks!
Pair the transpositions together. Each pair has one of each possible forms: $(12)(12),(12)(23),(12)(34)$. Notice that $(12)(12) = (123)(123)(123)$, $(12)(23) = (123)$, and $(12)(34) = (123)(234)$. Thus, altogether each permutation in $A_n$ is a product of $3$-cycles.