18.703 Modern Algebra, Cosets
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چکیده
Consider the group of integers Z under addition. Let H be the subgroup of even integers. Notice that if you take the elements of H and add one, then you get all the odd elements of Z. In fact if you take the elements of H and add any odd integer, then you get all the odd elements. On the other hand, every element of Z is either odd or even, and certainly not both (by convention zero is even and not odd), that is, we can partition the elements of Z into two sets, the evens and the odds, and one part of this partition is equal to the original subset H. Somewhat surprisingly this rather trivial example generalises to the case of an arbitrary group G and subgroup H, and in the case of finite groups imposes rather strong conditions on the size of a subgroup. To go further, we need to recall some basic facts abouts partitions and equivalence relations.
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