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.

 

[blue_raver22]

Hi Phoenix,

I'm glad you're taking the time to understand permutations and cycles in modern algebra. It can definitely be confusing at first, but with some practice, it will become second nature to you.

To answer your question, yes, η, θ, and other Greek letters have specific permutations depending on the number of cycles. In general, any permutation can be represented as a product of disjoint cycles, where each cycle represents a set of elements that are moved in a specific way. For example, a 3-cycle (1 2 3) represents a permutation that moves the element 1 to 2, 2 to 3, and 3 to 1.

In the case of θ and inverse θ, they will have the same cycle structure because they are inverse permutations of each other. This means that θ will have the same number of cycles as inverse θ, but the elements within each cycle will be reversed. For example, if θ has the cycle (1 2 3), then inverse θ will have the cycle (3 2 1).

I hope this helps clarify things for you. Keep practicing and you'll soon become comfortable with the notation and concepts. Good luck with your homework!

Best,