Permutations: Interpreting Image NotationI have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other.Let σ = ( 1 2 4 3 ), and τ = ( 1 3 2 4 ) in one-line notation.I thought σ is a '3, 4 swapping rearrangement of 1 2 3 4' and τ is a '2, 3 swapping rearrangement of 1 2 3 4' like swapping data in a computer memory.The problem is this : I interpreted σ τ as 'first swap 2 and 3, then you get 1 3 2 4, and then swap the elements in the 3rd and 4th place obtaining 1 3 4 2'.But the definition says σ τ = ( 1 4 2 3 ). I thought in reality, the permutation is just a place-changing operation, but by definition, composing two permutation is not just changing place twice. What is wrong in my interpretation?

Question

Permutations: Interpreting Image NotationI have a problem in interpreting permutation. I think the definition and my interpretation of it don&#039;t match each other.Let   σ  =  (  1     2     4     3  ), and   τ  =  (  1     3     2     4  ) in one-line notation.I thought   σ is a &#039;3, 4 swapping rearrangement of 1 2 3 4&#039; and τ is a &#039;2, 3 swapping rearrangement of 1 2 3 4&#039; like swapping data in a computer memory.The problem is this : I interpreted   σ  τ as &#039;first swap 2 and 3, then you get 1 3 2 4, and then swap the elements in the 3rd and 4th place obtaining 1 3 4 2&#039;.But the definition says   σ  τ  =  (  1     4     2     3  ). I thought in reality, the permutation is just a place-changing operation, but by definition, composing two permutation is not just changing place twice. What is wrong in my interpretation?

marktje28 · Accepted Answer

There are multiple levels of confusion, here (and I have deleted my previous answer because of this).Many posters here are accustomed to cycle notation, where   σ  =  (  1     2     4     3  ) means this:  σ  (  1  )  =  2    σ  (  2  )  =  4    σ  (  3  )  =  1    σ  (  4  )  =  3..It is also possible to employ &quot;image notation&quot; where   σ  =  (  1     2     4     3  ) means this:  σ  (  1  )  =  1    σ  (  2  )  =  2    σ  (  3  )  =  4    σ  (  4  )  =  3.Finally, sometimes στ means σ first, then τ, and sometimes it means τ first, then σ (as in ordinary function composition).So it&#039;s not immediately clear which conventions your text is using.A third source of confusion is what (1 2 4 3) is instructing us to do: are we swapping the NUMBERS 4 and 3, or whatever numbers are in the 3rd and 4th positions? This makes the whole situation even murkier.It appears your text is using &quot;image notation&quot;, and &quot;normal composition&quot;, so that:  σ  τ  (  1  )  =  σ  (  τ  (  1  )  )  =  σ  (  1  )  =  1  σ  τ  (  2  )  =  σ  (  τ  (  2  )  )  =  σ  (  3  )  =  4  σ  τ  (  3  )  =  σ  (  τ  (  3  )  )  =  σ  (  2  )  =  2  σ  τ  (  4  )  =  σ  (  τ  (  4  )  )  =  σ  (  4  )  =  3which is indeed (1 4 2 3) in &quot;image notation&quot;.

Permutations: Interpreting Image Notation I have a problem in interpreting permutation. I think the

Answered question

Answer & Explanation

New Questions in Pre-Algebra