Quotient groups I. Definition of quotient groups. Let G be a group and H a subgroup of G. Denote by G/H the set of distinct (left) cosets with respect to H. Quotient Groups. Let H be a normal subgroup of G. Then it can be verified that the cosets of G relative to H form a group. This group is called the quotient group or factor group of G relative to H and is denoted G/H. So the quotient group construction can be viewed as a generalization of modular . The definition of the quotient group uses cosets, but they are somewhat.


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Abstract algebra - Why the term and the concept of quotient group? - Mathematics Stack Exchange

quotient groups Sometimes quotient groups are forced upon us. We are not in possession of all the information. A simple example is the following. Assume that somebody is counting coins, but quotient groups only counting aid available to him is a light switch.

Every time he tallies one more coin he will toggle the light quotient groups This is a normal subgroup, because Z is abelian.

There are only two cosets: A slight generalization of the last example. Quotient groups again consider the group of integers Z under addition.

Let n be any positive integer. We will consider the subgroup nZ of Z consisting of all multiples of n.

Quotient group

Once again quotient groups is normal in Z because Z is abelian. The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.

  • Quotient group - Wikipedia
  • Undergraduate Mathematics/Quotient group
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When dividing 12 by 3 quotient groups obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects.

The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have quotient groups structure than an arbitrary collection of objects.


quotient groups These are the cosets of N in G. Because we started with a group quotient groups normal subgroup, the final quotient contains more information than just the number of cosets which is what regular division yieldsbut instead has a group structure itself.

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