Normal Subgroups and Quotient Groups
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چکیده
Another way to do this is to use individual elements. Take an element from {2, 6} and an element from {3, 7} and add them. Find the coset that contains the sum. That coset is the sum of the cosets. For example, if I use 6 from {2, 6} and 3 from {3, 7}, I get 6 + 3 = 1, which is in {1, 5}. Therefore, {2, 6}+ {3, 7} = {1, 5}. What happens if you choose different elements? Take 2 from {2, 6} and 7 from {3, 7}. Then 2 + 7 = 1, which is in {1, 5} again. Just as before, {2, 6}+ {3, 7} = {1, 5}. This is what it means to say that coset addition is well-defined: No matter which elements you choose from the two sets, the sum of the elements will always be in the same coset.
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