Solve Abstract Algebra Homework: Finding Left Coset (1,2,3)H with Permutations

[sportlover36]

One of my homework problems asks me to list the left coset (1,2,3)H where σ=(1,4,5)(2,3) and H=<σ>.

I know that you have to take the do the permutation of (1,2,3)(1,4,5)(2,3) but i am not sure how you can do that? I got (1,2,3)H={(1,2,3)(3)(1,2,4,5)} but i do not think that is right

 

[Dick]

H is the subgroup generated by sigma. It has more than one element. Try and figure out what H is first.

 

[sportlover36]

H is the subgroup generated by sigma. It has more than one element. Try and figure out what H is first.

it H={(1), (1,4,5)(2,3), (1,5,4)(2)(3), (1)(5)(4)(2,3), (1,4,5)(2)(3), (1,5,4)(2,3)}

 

[Dick]

it H={(1), (1,4,5)(2,3), (1,5,4)(2)(3), (1)(5)(4)(2,3), (1,4,5)(2)(3), (1,5,4)(2,3)}

Yes, that looks right. So the left coset (1,2,3)H will have six elements, right?

 

[sportlover36]

Yes, that looks right. So the left coset (1,2,3)H will have six elements, right?

which are {(1,2,3), (1,4,5,2,3), (1,5,4,2,3), (1,2,3), (1,4,5,2,3), (1,5,4,2,3)}...and i think i am doing it wrong :/

 

[Dick]

which are {(1,2,3), (1,4,5,2,3), (1,5,4,2,3), (1,2,3), (1,4,5,2,3), (1,5,4,2,3)}...and i think i am doing it wrong :/

I agree. Somethings going wrong. All of the elements of (1,2,3)H should be different. The fourth element of H is (2,3), right? (You don't have to write the (4)(5)). (1,2,3)(2,3) isn't (1,2,3).

 

[sportlover36]

I agree. Somethings going wrong. All of the elements of (1,2,3)H should be different. The fourth element of H is (2,3), right? (You don't have to write the (4)(5)). (1,2,3)(2,3) isn't (1,2,3).

umm is it (1,2,3)(2,3)= (1)(2,3)? I don't really know how to do ones that don't come out perfectly

 

[Dick]

umm is it (1,2,3)(2,3)= (1)(2,3)? I don't really know how to do ones that don't come out perfectly

Well, that's your problem. You should figure out how to multiply these cycles. Start with 1. (2,3) doesn't affect 1. (1,2,3) changes 1 into 2. So the product permutation must start with (1,2... Now where does 2 go?

 

[sportlover36]

Well, that's your problem. You should figure out how to multiply these cycles. Start with 1. (2,3) doesn't affect 1. (1,2,3) changes 1 into 2. So the product permutation must start with (1,2... Now where does 2 go?

ohh so (1,2,3)(2,3)= (1,2)(3) since 2 goes to 3 and 3 goes to 1

 

[Dick]

ohh so (1,2,3)(2,3)= (1,2)(3) since 2 goes to 3 and 3 goes to 1

Exactly, you step through the cycles from right to left and figure out where everything goes. It's pretty easy actually.

 

[sportlover36]

Exactly, you step through the cycles from right to left and figure out where everything goes. It's pretty easy actually.

so (1,2,3)(1,4,5)(2,3) = (1,4,5,2)(3)?

 

[Dick]

so (1,2,3)(1,4,5)(2,3) = (1,4,5,2)(3)?

Yes.