correspondence of normal subgroups and group congruences
We start with a definition.
It turns out that group congruences correspond to normal subgroups:
An equivalence relation is a group congruence if and only if there is a normal subgroup such that
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
|Title||correspondence of normal subgroups and group congruences|
|Date of creation||2013-03-22 15:32:52|
|Last modified on||2013-03-22 15:32:52|
|Last modified by||Dr_Absentius (537)|