correspondence of normal subgroups and group congruences
We start with a definition.
Definition 1.
Let be a group. An equivalence relation on is
called a group congruence
if it is compatible with the
group structure
, ie. when the following holds
-
•
-
•
So a group congruence is a http://planetmath.org/node/3403semigroup congruence
that additionally preserves the unary operation of taking inverse
.
It turns out that group congruences correspond to normal subgroups:
Theorem 2.
An equivalence relation is a group congruence if and only if there is a normal subgroup such that
Proof.
Let be a normal subgroup of and let be the
equivalence relation defines in . To see that this
equivalence relation is compatible with the group operation note
that if and then there are elements
and of such that and . Furthermore since is normal in there is an element
such that . Then we have
which gives that .
To prove the converse, assume that is an
equivalence relation compatible with the group operation and let
be the equivalence class
of the identity
. We will prove
that . We first prove
that is a normal subgroup of . Indeed if and then by the compatibility we have that , that
is ; so that is a subgroup of . Now if
and we have
Therefore is a normal subgroup of . Now consider two elements and of . To finish the proof observe that for we have
and
∎
Title | correspondence of normal subgroups and group congruences |
---|---|
Canonical name | CorrespondenceOfNormalSubgroupsAndGroupCongruences |
Date of creation | 2013-03-22 15:32:52 |
Last modified on | 2013-03-22 15:32:52 |
Owner | Dr_Absentius (537) |
Last modified by | Dr_Absentius (537) |
Numerical id | 7 |
Author | Dr_Absentius (537) |
Entry type | Theorem |
Classification | msc 20-00 |
Defines | group congruence |