# Abstract Nonsense

## Permutations (Pt. VI Alternate Definitions of the Sign of a Permutation cont.)

Point of post: This is a literal continuation of this post. I just hate when that dark grey to light grey thing happens. Just pretend that there was no lapse in the posts

And so we continue…

November 8, 2010

## Permutations (Pt. VI Alternate Definitions of the Sign of a Permutation)

Point of post: In this post we point out two different, but commonly used, definitions for the sign of a permutation and show that they’re equivalent to the one given here. Namely, the definitions given by the polynomial commonly denoted $\Delta(x_1,\cdots,x_n)$ in Lang (viz. ref. 1 pg. 63-64) and the definition by  inversions as in Shilov (viz. ref. 2 pg. 5-6 ) .

Motivation

Often in abstract and linear algebra books the definition of the sign of a permutation is not the same as the one given by me in the last post. This can be patently disturbing, since one will undoubtedly wonder “Are these the same definition, just said different?” and maybe more to the point “Why did they use this definition? Is it ‘better’, and if so, in what way?” I hope to answer these questions (to within the limitations of my time and the size of these posts) for two of the most commonly found definitions of the sign of a permutation.

November 8, 2010