Under what conditions does an equivalence relation induce a quotient that is a group?
Quotient group
Let be a group. Let be an equivalence
relation on the set . Let be the quotient map
induced by . Then is a group with
operation:
iff is a group homomorphism.
In turn, this is true of iff:
Background
The quotient of a set with respect to an equivalence
relation , expressed as , is simply the partition
implied by .
Given a group , it is natural to try and construct a
group using the set by inheriting the operation
from . To be inheritable, must
meet a certain requirement. The essence of the requirement is that two elements
of that map to the same equivalence class should be treated
alike in terms of . This requirement is what is made precise in the above definition.
Relation to cosets
The right and left side of the condition:
are satisfied by left and right cosets respectively, where the set in question, , is the equivalence class containing . In fact,
each of these 2 conditions, when paired with a subgroup is
sufficient to define an equivalence relation on .
To see the connection, consider a right coset, , and consider two elements of the coset: the founding and another arbitrary element, . Notice, by property of right cosets, that can be expressed as for some . So ,
but as is equivalent to , must be in the same equivalence class as .
Connection to the and
These two forms can be understood as expressing the fact that elements within an equivalence class act the same at the granularity of equivalences classes which is necessary for the quotient to be a group. Below is a visualization of this idea.
Alternative notation
The group operation is often defined compactly as: