Quotient group

The factor group or quotient group is a group that is formed by a standard construction from a given group with the aid of a normal subgroup. It is called with.

Construction

The elements of this case are the cosets with respect to, ie

Finally you need an inner join, namely

Can be obtained from the normal subgroup property of showing that a group is called the group of factor G to N, and that corresponds to the complex product. Is the neutral element of and to the inverse element is given by. The order of the factor group of named and designated in the index. Is a finite group, the following applies.

Conversely, one can show that a subgroup of a group is a normal subgroup, if the equality holds for all.

In Abelian groups, each subgroup is a normal subgroup. Thus, the factor group can form there by each sub-group, which in turn is then abelian.

Example

Residue class group of the additive group of integers

The group is an abelian group. For each is a subgroup, and in particular a normal subgroup of. The factor group is called a residue class group modulo. She has exactly elements and together with a multiplicative combination of the so-called residue class ring.

Be your elements as

Written and hot congruence with respect to addition modulo. It is therefore

The internal linking is usually re- designated. In true, for example

There, so.

Factor group by nuclei of homomorphisms

In case of a group homomorphism between two groups, and is the core of a normal subgroup of and the image of a subset of. Therefore, a factor group. The homomorphism theorem for groups states that this factor group is isomorphic to the image of.

Universal property of the quotient group

The figure with with core is an epimorphism, ie a surjective homomorphism. The universal property states that, for every group homomorphism with exactly one homomorphism exists with.

Example: Let the natural projection of the integers in the residue class group modulo 6 Be homomorphism. Then lies at the core of, and is given by:

.

This left are the residue classes modulo 6 and right the residue classes modulo 3