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 .
Secondly, we can calculate
whence the associativity is in in the groupoid .
Let be the identity element of and . Then
and therefore is an identity element in .
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:
These apparently form the cyclic group of order 3.
|Title||homomorphic image of group|
|Date of creation||2013-03-22 18:56:27|
|Last modified on||2013-03-22 18:56:27|
|Last modified by||pahio (2872)|