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## Homework Statement

Let n>=2 n is natural and set x=(1,2,3,...,n) and y=(1,2). Show that Sym(n)=<x,y>

## Homework Equations

## The Attempt at a Solution

Approach: Induction

Proof:

Base case n=2

x=(1,2)

y=(1,2)

Sym(2)={Id,(1,2)}

(1,2)=x and Id=xy

so base case holds

Inductive step assume Sym(k)=<x,y> where x=(1,2,3,4,...,k) and y=(1,2)

Show Sym(k+1)=<t,y> where t={1,2,...,k+1} and y=<1,2>

To prove the inductive step, I was thinking we have to use the fact that every b in Sym(k+1) can be represented as the product of pairwise disjoint cycles. I think that we can express $\lambda$ in terms of x