Modern Algebra: Permutations and Cycles

[Phoenixtears]

Hi there,

I'm doing homework right now (no this isn't a homework question!) and have basic questions on permutations and cycles. The concept seemed so simple in class and still seems simple, but the notation using lowercase Greek letters is confusing me.

Do η and \theta and most of the other Greek letters (we went from \alpha to \gamma in class) have specific permutations, depending on the number of cycles?

For example, one problem involves showing that \theta and inverse\theta have the same cycle structure, but I can't figure out how to represent theta because I don't know it's specific disjoint cycles or transposition.

Thank you so very much for the help!

Phoenix

 

[rasmhop]

Usually what is meant is for you to show this for arbitrary permutations. You need to show that \theta and \theta^{-1} have the same cycle structure no matter what permutation \theta is. Some ideas for this may be gained by considering specific examples like
(1 2 3)
(1 2)(3 4)
etc.
and seeing how their inverses look, but you are asked to do it in general.

 

[Phoenixtears]

Ah, that makes perfect sense to me. Thank you so much!

The examples we did in class involved specific cycle structures for alpha to gamma, and I didn't quite understand how those came to be. What does the inverse of a permutation represent, then, if there is no specific cycle structure?

 

[Deveno]

one of the theorems you should have learned (or maybe will be learning soon), is that every permutation can be written as a product of (disjoint) cycles.

so understanding cycles is a big part of understanding permutations, in general.

and for cycles, there is a nifty trick, which it pays to remember:

if θ = (a b c ... k)

then θ^-1 = (k ... c b a) = (a k ... c b)

(just "mirror" the original cycle).

the reason that greek letters are usually used for permutations, is that roman letters are often used as symbols to stand for the (perhaps unknown or arbitrary) set elements that the permutations permute (boy...that's a mouthful).

so α(a) will be properly understood as the image under α of a, and not confused as, say, the composition of two permutations.

in other words, there's two layers going on:

the set layer <--> elements (roman letters)
the group layer <--> permutations (functions) (greek letters).

there is no "convention" for associating a particular cycle type with a particular greek letter, although τ (tau) is often used for transpositions (2-cycles), perhaps "T" for "transposition"?

another common letter used for an arbitrary permutation is π (pi) ("P" for permutation?), which confuses a lot of people, since they are used to pi being a number.

personally, i like σ (sigma) and μ (mu), as they are easy to distinguish from typical roman letters (α (alpha) and β (beta) can get mixed up with a and b, and γ (gamma) looks too much like y).

to prove θ and θ^-1 have the same cycle-type (decomposition into disjoint cycles), it helps to know (which is another theorem/lemma that should be proved in your book/class) that disjoint cycles commute.