Question

Prove that every elements of$A_n$is a product of n-cycles

Short answer

It is proved that every elements of$A_n$ is a product of n-cycles.

Step-by-step solution

Exercise 30

From exercise 30, we have every element of $A_n$is a product of 3-cycles. So,now we need to show that all the 3-cycles are product of -cycles.

Prove the result

Assume that be any 3-cycles and we have the set ,$\left\{d_1,d_2,\dots ,d_{k-3}\right\}=\left\{1,2,\dots ,n\right\}-\left\{a,b,c\right\}$ this implies that we have:

$\left(bac{d}_1\dots d_{k-3}\right)\left(d_{k-3}\dots d_{1}cba\right)=\left(abc\right)$

This implies that$\left(abc\right)$ is a product of two n-cycles.

Hence, it is proved that every elements of $A_n$is a product of n-cycles