What are Cosets? - Understanding & Solving Problems

[semidevil]

so I'm solving problems that tell me to find the left cosets, but I don't really know what they are.

by defn, let G be a group and H a subgp of G.. and let a be an element of G. the set ah for any h in H, denoted by aH is the left coset.

I mean, what does that mean. so for an example problem. find the left cosets of {1, 11} in U(30). So U(30) has order 8, with elements 1 7 11 13 17 19 23 29. By formula, order of G/H equalis the number of left cosets. so 8/2 = 4. meaning we have 4 left cosets. and the book says the cosets are H 7H 13H and 19H.


so exactly why? what are those numbers? how did they derive that?

at first, I thought you just take each element and multiply by H, , so aH = 1H, 3H, 7H...29H,but I guess I was way off.

 

[Janitor]

at first, I thought you just take each element and multiply by H, , so aH = 1H, 3H, 7H...29H,but I guess I was way off.

I believe that is what you in fact do. But what you should find from using the multiplication table for G is that you get a second repetition of the same four cosets, i.e. you really only have four distinct cosets, not eight.

 

[semidevil]

I believe that is what you in fact do. But what you should find from using the multiplication table for G is that you get a second repetition of the same four cosets, i.e. you really only have four distinct cosets, not eight.

ok, so if I do the multiplication table:

1 * {1, 11} = {(1*1) (1*11)}
7 * {1, 11} = {7*1), (7*11)}
.
.
.
.
.
29 *{1, 11} = {29*1) (29*11)}

and that mod 30,

I get
1, 11
7, 17
11, 1
13, 23
17, 7
19, 29
23, 13
29, 19

I dotn know where the 4 distinct cosets come from

 

[Janitor]

To emphasize that your eight rows of pairs are eight sets (which is what a coset is, after all), write them with brackets:

{1, 11}
{7, 17}
{11, 1}
{13, 23}
{17, 7}
{19, 29}
{23, 13}
{29, 19}

Remember that the order that you list the elements in a set doesn't matter; it's the same set. So {1, 11} is the same set as {11, 1}, and so on. So throw out four redundant sets from your list of eight, leaving you with four distinct sets. You were 99% of the way done with the problem where you left off.