Let a and b belong to Sn. Prove that (a^-1)(b^-1)(a)(b) is an even permutation.

[tyrannical608]

Homework Statement

Let a and b belong to Sn. Prove that (a^-1)(b^-1)(a)(b) is an even permutation.

Homework Equations

Definitions I have are
Every permutation in Sn, n>1 is a product of 2 cycles
and
A permutation that can be expressed as a product of an even number of 2 cycles is called an even permutation

Thanks

 

[HallsofIvy]

So, it appears you are saying that all the permutations involved in a^{-1}b^{-1}ab is a two-cycle and this is a product of 4 of them!

 

[tyrannical608]

So, it appears you are saying that all the permutations involved in a^{-1}b^{-1}ab is a two-cycle and this is a product of 4 of them!

Maybe its my flu...but what do you mean exactly? What I said are two definitions out of the book.

Do I have to show that a^-1b^-1 is one cycle and that ab is another cycle?

 

[AKG]

If a can be expressed as a product of k 2-cycles, what can you say about how many 2-cycles it takes to express a^-1?