Determining parity of permutations

May 2010
4
0
South Africa
Hello, i have an exam coming up in 2 days...

I came across this question in my notes which I dont understand.

there is an attachment to this post for the questions (186 and 187) which confuse me

I dont understand what they mean by parity and what they mean by S_5

anybody know how to tackle this?
 

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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Every permutation can be written as a succession of "transpositions" where a transposition means swapping just two elements.

For example, the permutation \(\displaystyle \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{pmatrix}\) can be done by : first swap 1 and 3 to get \(\displaystyle \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 4 & 4\end{pmatrix}\), then swap 4 and 5 to get the final result. Now, you can often write a permutation as a series of "swaps" in different ways. For example, here I could say "first swap 1 and 2, then swap 2 and 4 to get 41325, then swap 4 and 5 to get 51324, then swap 2 and 5 to get 21354, then swap 2 and 3 to get 31254, and finally, swap 1 and 2 to get 32154, the same result as if we had just swapped 1 and 3 and then 5 and 4.

Or course, that is a lot more complicated because I did a lot of uneccessary swaps- but notice that for each "unecessary swap" I had to undo that swap- that mean I added pairs of swaps so that instead of only 2 swaps I did 6- four more swaps. In fact, all "unecessary swaps" come in pairs so we have: if one way of writing a permutation in terms of transpositions requires an even number of transpostions, the any way of writing that permutation as a sequence of transpositions must use an even number of transpostions. Similarly, for odd numbers of transpositions.

That means that every permutation can be characterized as an "even" permutation (if it can be written as a sequence of an even number of transpostions) or as an "odd" permutation (if it can be written as a sequence of an odd number of transpositions). That is the "parity" of a permutation- even or odd.

"\(\displaystyle S_5\)" simply means "the set of all permutations on 5 objects".