Let be a group and let be a subgroup of . Using the definitions of the sets , , and , prove that for every in if and only if .

= { in | }

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- Oct 5th 2008, 11:48 AMll2008cosets
Let be a group and let be a subgroup of . Using the definitions of the sets , , and , prove that for every in if and only if .

= { in | } - Oct 5th 2008, 08:33 PMThePerfectHacker
- Oct 5th 2008, 10:06 PMll2008
I thought about that, but I don't think I can use those theorems to prove it. I would have to prove it using the definitions.

- Oct 8th 2008, 12:23 AMgosualite
Hacker's way is valid. It's using the definitions, not theorems. Here it is essentially broken down into baby steps.

If , then implies , whence .

If for all , it follows that (and so equality too of course).