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## Homework Statement

I am working through MacLane/Birkhoff's Algebra, and in the section on Symmetric and Alternating groups, the last few exercises deal with generators and Defining relations for Sn (the symmetric group of degree n). These read:

11. Prove that Sn is generated by the cycles (1 2 ... n-1) and (n-1 n).

12. In Exercise 11 determine defining relations on these two generators.

13. Prove that Sn is generated by the transpositions (1 2), (2 3), ..., (n-1 n).

14. In Exercise 13 show that the generators t[itex]_{i}[/itex] = (i i+1), i =1,...,n-1 satisfy the defining relations

(t[itex]_{i}[/itex])[itex]^{2}[/itex] = 1

t[itex]_{i}[/itex]t[itex]_{j}[/itex] = t[itex]_{j}[/itex]t[itex]_{i}[/itex] if i-j≠±1

and

(t[itex]_{i}[/itex]t[itex]_{i+1}[/itex])[itex]^{3}[/itex] = 1

and that this is a complete list of relations.

## Homework Equations

Thm: Any permutation on n is a composite of transpositions.

## The Attempt at a Solution

I believe I understand why/how the given generators do generate Sn in problem 11:

any transposition of elements (j k) can be constructed using the cycles x = (1 2 ... n-1) and y = (n-1 n) as (x[itex]^{-j}[/itex]°y°x[itex]^{j}[/itex])°(x[itex]^{-k}[/itex]°y°x[itex]^{k}[/itex])°(x[itex]^{-j}[/itex]°y°x[itex]^{j}[/itex])

And by the theorem, we can represent any permutation (any element of Sn) as a composition of these transpositions.

and in problem 12:

the transposition (1 2 ... n-1) can be written as (n-1 n-2)...(4 3)°(3 2)°(2 1) and (n-1 n) is already of the given form, so the given transpositions can be combined to form the permutations of 11 which in turn generate Sn.

What I am having difficulty with is what the process is for determining defining relations and then showing that they form a complete list of relations. Perhaps showing the order of each of the generators (x[itex]^{n-1}[/itex]=1 and y[itex]^{2}[/itex]=1) and how the generators "commute" with one another is standard? If so how in the world do you figure out what these additional relations are?

Thank you all, this is my first post, so feedback on the amount/quality of information given is welcome and appreciated.

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