What Are the Left Cosets of H in S_3?

[BustedBreaks]

I am having trouble understanding this example:

Let G=S_3 and H={(1),(13)}. Then the left cosets of H in G are

(1)H=H
(12)H={(12), (12)(13)}={(12),(132)}=(132)HI cannot figure out how to produce this relation:

(12)H={(12), (12)(13)}={(12),(132)}=(132)H

I understand (12)H={(12), (12)(13)} but not how (12)(13) = (132) or the equivalence after that...

 

[daveyinaz]

Note that since (12) and (13) are in S3, they are actually the elements (12)(3) and (13)(2), where 3 and 2 are fixed respectively.

(12)(13), we do computation from right to left, so 1 goes to 3 from (13), then 3 goes to itself in (12). So we have (13...). Now 3 goes to 1 in (13) and 1 goes to 2 in (12), so in (13...) we have 3 goes to 2. Hence (132). This is a quick dirty answer to your question, but I think you should reread or review computations done using permutations.

 

[BustedBreaks]

Note that since (12) and (13) are in S3, they are actually the elements (12)(3) and (13)(2), where 3 and 2 are fixed respectively.

(12)(13), we do computation from right to left, so 1 goes to 3 from (13), then 3 goes to itself in (12). So we have (13...). Now 3 goes to 1 in (13) and 1 goes to 2 in (12), so in (13...) we have 3 goes to 2. Hence (132). This is a quick dirty answer to your question, but I think you should reread or review computations done using permutations.

Okay so I get your method here and I am trying to apply it to this one (23)(13) but I am not getting the answer the book has which is (123)

I set it up like this

(23) (13)
123 123
132 321

then

so 1 goes to 3 then 3 goes to 2, 2 goes to 2 then 2 goes to 3, 3 goes to 1 and 1 goes to 1 so I get 231... I know you said your way was quick and dirty, so maybe I am missing something completely?

EDIT:

Okay so I'm pretty sure that 231 and 123 are the same thing but is there a preference for writing it out?

 

[mrbohn1]

The convention is to start with the smallest number.