homomorphic image of group
Theorem. The homomorphic image of a group is a group. More detailed, if is a homomorphism from the group to the groupoid , then the groupoid also is a group. Especially, the isomorphic image of a group is a group.
Proof. Let be arbitrary elements of the image and some elements of such that . Then
whence is closed under “”, and we, in fact, can speak of a groupoid .
Accordingly, is a group.
Remark 1. If is Abelian, the same is true for .
Remark 2. Analogically, one may prove that the homomorphic image of a ring is a ring.
Example. If we define the mapping from the group to the groupoid by
then is homomorphism:
The image consists of powers of the residue class (http://planetmath.org/Congruences) , which are
These apparently form the cyclic group of order 3.
Title | homomorphic image of group |
---|---|
Canonical name | HomomorphicImageOfGroup |
Date of creation | 2013-03-22 18:56:27 |
Last modified on | 2013-03-22 18:56:27 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 14 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 20A05 |
Classification | msc 08A05 |
Related topic | GroupHomomorphism |
Related topic | CorrespondenceBetweenNormalSubgroupsAndHomomorphicImages |