Proof about symmetric groups and generators

  • #1

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
 

Answers and Replies

  • #2
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Hint: ##(1\;2\; \dots \; k\; (k+1)) =(1\;2\; \dots \; k)\circ (k \; (k+1))## - from right to left. So ##(k \; (k+1)) = (1\;2\; \dots \; k)^{-1} \circ (1\;2\; \dots \; k\; (k+1)) \in \mathcal{Sym}(k+1)## by induction and the fact that all permutations that doesn't involve ##k+1## are in
If you can show, that all other transpositions ##(i \; (k+1))## are in it, too, you're done. This is because you then can place ##k+1## in every position with an arbitrary fixed order of the other elements.
Remember that all transpositions ##(i \; j)## with ##i\, , \, j \, \leq \, k## are already in the group.

Edit: Plus the fact that ##\{ \pi \in \mathcal{Sym}(k+1) \; | \; \pi(k+1) = k+1 \} \cong \mathcal{Sym}(k) \subset \mathcal{Sym}(k+1)##
 
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