Transcription of Cosets, Lagrange’s theorem and normal subgroups
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Cosets, Lagrange’s theorem and normal subgroups
Cosets, Lagrange s theorem and normal subgroups1 CosetsOur goal will be to generalize the construction of the groupZ/nZ. Theidea there was to start with the groupZand the subgroupnZ= n , wheren N, and to construct a setZ/nZwhich then turned out to be a group(under addition) as well. (There are two binary operations + and onZ, butZis just a group under addition. Thus, the fact that we can alsodefine multiplication onZ/nZwill not play a role here, but its naturalgeneralization is very important in Modern Algebra II.) We would like togeneralize the above constructions, beginning with congruence modn, tothe case of a general groupG(written multiplicatively) together with asubgroupHofG.
equivalence relation ‘ (mod H), is denoted G=H. Right cosets Hg= fhg: h2Hgare similarly de ned. They are equiva-lence relations for the equivalence relation r (mod H) de ned by: g 1 rg 2 (mod H) if g 2g 1 1 2H, or equivalently if there exists an h2Hsuch that g 2g 1 1 = h, i.e. if g 2 = hg 1 for some h2H. The set of all equivalence classes
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