Using induction on n that if n is odd, then an n-cycle can be written as a product of 3-cycles.

Question

Here n is greater than or equal to 4 and we are working in Sn.

The hint states that given an n-cycle delta, find a 3-cycle tau such that the product delta*tao is an (n-2) cycle.

I am having trouble finding tao that makes the n cycle a product of 3-cycles.

Answer 1

IF the question has the condition that $n$ is odd (which is stated in the title of your question but not explicitly in your question) then what does this tell you about $\delta$ in terms of being odd or even. Then think about generating sets for $A_{n}$. If you have not done any generating sets for $A_{n}$ then think about just 2 transpositions and combining them as a single cycle. Be careful to consider all possibilities of the 2 transpositions.