Question

Let S_n denote the group of all permutations of the set [n]={1,2,...,n}. A permutation sigma in S_n is called an involution if sigma^2 =1, where 1=(1) in S_n denotes the identity. Explicitly list all evolutions of S_1, S_2, S_3, S_4. Fore general n, show that a permutation 1 not equal to sigma in S_n is an involution if and only if sigma is a product of disjoint 2-cycles

Answer 1

How do you feel about part a?

Answer 2

S_n are sufficiently simple for \(n\in\{1,\dots,4\}\) to manually list the involutions. an involution is essentially just a permutation that is its own inverse, i.e. it undoes itself when applied a second time

Answer 3

proving that involutions in S_n are precisely products of disjoint 2-cycles is easy enough in the product of disjoint 2-cycles => involution direction, since they tell you that all 2-cycles are involutions. it is slightly less easy to show the other direction, namely that an arbitrary involution is a product of disjoint 2-cycles

Answer 4

Perhaps you can begin by considering for each of the sets of permutations S_1, S_2, S_3, S_4. \[ [1] = \{1\} \\ [2] = \{1,2\} \\ [3] = \{1,2,3\} \\ [4] = \{1,2,3,4\}.\] Then \[S_1 = \{1\} \\ S_2 = \{1,(1,2)\} \\ \vdots\]

Answer 5

I do not fully know how to complete part b

Answer 6

Do you recognize that disjoint cycles commute with one another? Suppose then we have a permutation \(\sigma\) that can be factored as a product of disjoint 2-cycles \(\sigma_1\cdots\sigma_n\). It follows then that $$\sigma^2=\sigma_1\cdots\sigma_n\sigma_1\cdots\sigma_n=\sigma_1^2\dots\sigma_n^2=e^n=e$$since 2-cycles are obviously involutions, i.e. \(\sigma_1^2=\sigma_2^2=\dots=\sigma_n^2=e\), the identity; since they are disjoint, we were able to commute the \(\sigma_1,\dots,\sigma_n\) so that they square to the identity

Answer 7

Okay, to find the converse do I assume a disjoint cycle and show that it is an involution?

Answer 8

yes, exactly

Answer 9

I am sorry but I am still confused with what you have written. Would you mind explaining it to me. Is that just one way or both ways of the proof.

Answer 10

that's only one way

Answer 11

Okay. SO that is saying that we have an involution sigma^2=(1) and leading to disjoint 2 cycles?

Answer 12

maybe it would help if you did it the other way too. S
o I can see the entire picture

Answer 13

that is saying that a product of disjoint 2-cycles is an involution. to go the other way, suppose we have some permutation \(\sigma\). by cycle decomposition there is a more-or-less (i.e. up to commutativity) unique representation in products \(\sigma=\sigma_1\cdots\sigma_n\). if it's an involution we have \(\sigma^2=e\implies\sigma=\sigma^{-1}\) so $$\sigma_1\cdots\sigma_n=\sigma_n^{-1}\cdots\sigma_1^{-1}$$note the cycles are disjoint we can commute freely. do you see a way to deduce that \(\sigma_i=\sigma_i^{-1}\)

Answer 14

That makes more sense now. Thanks a ton!