Quotient group
Group obtained by aggregating similar elements of a larger group / From Wikipedia, the free encyclopedia
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A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory.
For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written , where is the original group and is the normal subgroup. (This is pronounced , where is short for modulo.)
Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of . Specifically, the image of under a homomorphism is isomorphic to where denotes the kernel of .
The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.
Given a group and a subgroup , and a fixed element , one can consider the corresponding left coset: . Cosets are a natural class of subsets of a group; for example consider the abelian group G of integers, with operation defined by the usual addition, and the subgroup of even integers. Then there are exactly two cosets: , which are the even integers, and , which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).
For a general subgroup , it is desirable to define a compatible group operation on the set of all possible cosets, . This is possible exactly when is a normal subgroup, see below. A subgroup of a group is normal if and only if the coset equality holds for all . A normal subgroup of is denoted .
Definition
Let be a normal subgroup of a group . Define the set to be the set of all left cosets of in . That is, .
Since the identity element , . Define a binary operation on the set of cosets, , as follows. For each and in , the product of and , , is . This works only because does not depend on the choice of the representatives, and , of each left coset, and . To prove this, suppose and for some . Then
- .
This depends on the fact that is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on .
To show that it is necessary, consider that for a subgroup of , we have been given that the operation is well defined. That is, for all and for .
Let and . Since , we have .
Now, and .
Hence is a normal subgroup of .
It can also be checked that this operation on is always associative, has identity element , and the inverse of element can always be represented by . Therefore, the set together with the operation defined by forms a group, the quotient group of by .
Due to the normality of , the left cosets and right cosets of in are the same, and so, could have been defined to be the set of right cosets of in .
Example: Addition modulo 6
For example, consider the group with addition modulo 6: . Consider the subgroup , which is normal because is abelian. Then the set of (left) cosets is of size three:
- .
The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.
The reason is called a quotient group comes from division of integers. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects.[citation needed]
To elaborate, when looking at with a normal subgroup of , the group structure is used to form a natural "regrouping". These are the cosets of in . Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.