When is a normal subgroup of a group the quotient group is obtained by "collapsing the elements of to the identity." More precisely, the set is defined as the set of equivalence classes where two elements are considered equivalent if the cosets and are the same.
By far the most well-known example is where is some positive integer and the group operation is addition. Then is the additive group of integers modulo So the quotient group construction can be viewed as a generalization of modular arithmetic to arbitrary groups. In fact, the quotient group is read " mod "
The quotient is a well-defined set even when is not normal.
Let be a group and a subgroup. Then is the set of left cosets as runs over the elements ofThis set is used in the proof of Lagrange's theorem, for instance. In fact, the proof of Lagrange's theorem establishes that if is finite, then Note that
If is normal, then the set has a natural group structure; because This gives a formula for multiplying cosets. Another way to express this formula is as follows:
If is a normal subgroup of then the function given by is a group homomorphism.
The definition of the quotient group uses cosets, but they are somewhat unwieldy to work with. It is often easier to denote the coset by the notation ; then as expected. The important point is that this is true no matter which representatives are chosen: if and then so Another way to say this is as follows: the coset containing the product of two coset representatives is independent of the choice of representatives.
This is not true if is not normal.
Let the symmetric group on three symbols. Let be the two-element subgroup generated by the transposition Then consists of three cosets:
Since is not normal, it does not inherit the group structure from ; in other words, the product of two coset representatives will land in a coset that is not independent of the choice of representatives.
For example, taking and the product of these two representatives is which is in But instead taking and we find .
When (with group law given by addition) and the quotient is the set of cosets of The coset representatives that are usually chosen are . So for instance the coset is abbreviated and The addition in is as expected: If subtracting will give the coset representative in the range For example, Here because so
Main article: Group isomorphism theorems
The three fundamental isomorphism theorems all involve quotient groups. The most important and basic is the first isomorphism theorem; the second and third theorems essentially follow from the first. Here are some examples of the theorem in use.
(First Isomorphism Theorem) A group homomorphism induces an isomorphism defined naturally by
The complex numbers such that form a group under multiplication. Call this group (for unit circle). Show that where and denote the additive groups of real numbers and integers, respectively.
Consider the function given by Then is clearly surjective, because every complex number with absolute value can be written as for some real number (by Euler's formula). The kernel of is the set of real numbers such that i.e. and This happens if and only if is an integer, so
The result follows directly from the first isomorphism theorem.
Another example of the first isomorphism theorem is an appealingly nontrivial example of a non-abelian group and its quotient.
Consider the symmetric group on four symbols. It permutes the vertices of this tetrahedron: There are three pairs of disjoint edges: the two purple edges, the blue/green pair, and the red/yellow pair. Any permutation of the vertices will permute the edges in such a way as to move these pairs onto each other. For instance, a transposition of the red and yellow vertices will fix the purple edge pair, but the red/yellow pair will swap places with the blue/green pair.
So any permutation in will have an associated permutation of these three objects (the edge pairs). This gives a function It is a homomorphism (essentially tautologically, since the group operation on both sides is just composition of functions). Its kernel has four elements in it, consisting of the identity and the three double transpositions. (Any double transposition will fix both edges in one pair, and will swap the edges in the other two pairs. For example, swapping the yellow and red vertices and then swapping the blue and green vertices will leave the purple edge pair unchanged, but will swap the blue and green edges, and the yellow and red edges. The three pairs stay in the same place, even though the two edges in some pairs may have switched places.) And it is not hard to show that is surjective.
So the first isomorphism theorem gives an isomorphism
Let be a group, and a normal subgroup. Which of the following statements is/are always true?
I. If is finite and is finite, then is finite.
II. If is finite and cyclic and is finite and cyclic, then is finite and cyclic.
III. If is abelian and is abelian, then is abelian.
- A finite cyclic group is a group that is isomorphic to the integers mod for some
- An abelian group is a group whose operation is commutative: for all
- Atiliogomes, . Adjacent-vertex-distinguishing total coloring of complete graph K4. Retrieved May 25, 2013, from https://commons.wikimedia.org/wiki/File:Avd-total-coloring-of-complete-graph-K4.svg