0

I'm a bit confused on the use of cycle notation

on this website https://groupprops.subwiki.org/wiki/Element_structure_of_symmetric_group:S4

it has the identity relation () in one line notation as (1 2 3 4)

But doesn't (1 2 3 4) imply that 1 maps to 2, that 2 maps to 3 and that 3 maps to 4 and 4 maps to 1?

How does that signify the identity relation?

Temirzhan
  • 983
  • 1
  • 11
  • 25
  • 1
    The one line notation is not the cycle notation. Look at the chart on that web page labeled "multiple ways..." for an explanation. The one line notation is in effect the graph, or a tabular representation of the function represented by the permutation. I mean, perm $\pi$ has one line notation $(\pi(1),\pi(2),\pi(3),\pi(4))$, so if $\pi$ is the identity ($pi(x)=x$ for all $x$ ) you get the notation $(1,2,3,4)$. – kimchi lover Sep 26 '17 at 01:01

1 Answers1

1

The one-line notation (without parentheses!) is a compact notation for the more traditional two-line notation: \begin{alignat}{2} \text{one-line notation}&&&\text{cycle notation}\\ \begin{matrix}1&2&3&4\end{matrix}&\longleftrightarrow\begin{pmatrix}1&2&3&4\\1&2&3&4\end{pmatrix}& {}\longleftrightarrow{}&()\\ \begin{matrix}1&2&4&3\end{matrix}&\longleftrightarrow \begin{pmatrix}1&2&3&4\\1&2&4&3\end{pmatrix}& {}\longleftrightarrow{}& (34)\\ \begin{matrix}4&3&2&1\end{matrix}&\longleftrightarrow \begin{pmatrix}1&2&3&4\\4&3&2&1\end{pmatrix}& {}\longleftrightarrow{}&(14)(23) \end{alignat}

Bernard
  • 175,478
  • one line notation directly to cycle notation say I wanted to go from (4 3 2 1) into cycle notation? how could that be done? – Temirzhan Sep 26 '17 at 01:23
  • Where does 1 go? In position 1 of 4321 is a 4, so it goes to four. So far, the cycle notation is (14. Where does 4 go? In position 4 is a 1. So 4 goes back to one, and the cycle is (14). Where does 2 go? In position two is a 3, and in position 3 is a 2, so the next cycle is (23). In cycle notation, (14)(23). – Kyle Miller Sep 26 '17 at 02:50
  • @Aldrec: I agree with the above comment, but you see this means you need, at least implicitly, the two-line notation. Actually, I think one cannot do without the intermediate step. – Bernard Sep 26 '17 at 09:28