QuotientGroups - Quotient Group Examples 1. The basic...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

Page1 / 3

QuotientGroups - Quotient Group Examples 1. The basic...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online