Math Help - unable to understand cosets

The definition of coset is as follows



However, I am still not able to understand what this means. An example shown is





I do not understand how 3 + H = {3, 1}. 3 does not belong in H. and I also don't understand 2+H = {2, 0}.

I can't make out much from the textbook. Where can I learn this material? I would like to see some more simplified and friendly defintion. In fact for past couple of hours I am struggling to get a grasp on cosets.

Another example would be



i don't understand it

Originally Posted by xyz

I do not understand how 3 + H = {3, 1}. 3 does not belong in H. and I also don't understand 2+H = {2, 0}.

I can't make out much from the textbook. Where can I learn this material? I would like to see some more simplified and friendly defintion. In fact for past couple of hours I am struggling to get a grasp on cosets.

It says H = {0,2}

so adding the element 1 to subgroup H to make a left co-set gives 1+H = {1,3} as one is added to each element of H e.g {0+1,2+1} = {1,3}

as follows 2+H = 2+{0,2} = {0+2,2+2} = {2,0}

Does this help?

Okay, its supposed to be 2+H = 2+{0,2} = {0+2,2+2} = {2,0}. However, isn't 2+2 = 4? so shouldn't it be {2, 4}?

and 4=0 because 4 and 0 represent the same element in Z_4 Group.