Gruppe G

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Gruppe G

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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.

When dividing 12 by 3 one 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 more structure than an arbitrary collection of objects.

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

Consider the group of integers Z under addition and the subgroup 2 Z consisting of all even integers. This is a normal subgroup, because Z is abelian.

A slight generalization of the last example. Once again consider the group of integers Z under addition. Let n be any positive integer.

We will consider the subgroup n Z of Z consisting of all multiples of n. Once again n Z is normal in Z because Z is abelian. This is a cyclic group of order n.

The twelfth roots of unity , which are points on the complex unit circle , form a multiplicative abelian group G , shown on the picture on the right as colored balls with the number at each point giving its complex argument.

Consider its subgroup N made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue.

One can check that the cosets form a group of three elements the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.

Consider the group of real numbers R under addition, and the subgroup Z of integers. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1.

The group N is known as the special linear group SL 3. The Paillier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of G without knowing the factorization of n.

If G is finite, the index is also equal to the order of G divided by the order of N. Its kernel is N. Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.

This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups.

This is a special case of the extension problem. The result is known as a homogeneous space. From Wikipedia, the free encyclopedia. Subgroup Normal subgroup Quotient group Semi- direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics.

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Artin, AlgebraProposition 6. Contact Us Privacy Policy Menu. In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds Tisch Mexico, regular actions of algebraic groups on algebraic varietiesand actions of group schemes Bgo Com schemes. Explore more.
Gruppe G
Gruppe G
Gruppe G

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2 Comments

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