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Unformatted text preview: Quotient Group Examples 1. The basic example: Z /n Z . Let Z = ( Z , +) and n Z = ( n Z , +). You should verify n Z Z . Since Z is abelian, we get n Z C Z for free. A typical coset is a + n Z for a Z . Z /n Z = { a + n Z : a Z } = { i + n Z : 0 i < n } = Z n (the latter notation we will abandon because of its ambiguity with other notation in abstract algebra). The group operation in Z /n Z is ( a + n Z )+( b + n Z ) = ( a + b )+ n Z = r + n Z where ( a + b ) = qn + r and 0 r < n . Thus the operation in Z /n Z is identical to the modular addition that we were performing in Z n . e.g. Z / 3 Z = { 0 + 3 Z , 1 + 3 Z , 2 + 3 Z } . (note that Z / 3 Z has only 3 distinct cosets. it is convenient for us to use 0, 1 and 2 as the coset representatives for each of these cosets, but we could just as easily write Z / 3 Z = { 21 + 3 Z , 2 + 3 Z , 14 + 3 Z } . Once again we listed the exact same 3 distinct cosets of Z / 3 Z . 21+3 Z = 0+3 Z , 2+3 Z = 1+3 Z , 14+3 Z = 2+3 Z ....
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 Spring '11
 Johnson
 Algebra

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