Coset

1 Basic definition

A coset is a subset of a group obtained by combining each element of a subgroup with a fixed element from the larger group. The idea captures how a group can be divided into translated copies of the same smaller structure. Cosets are central in group theory because they connect subgroup structure with counting arguments, equivalence relations, and quotient constructions.

1.1 Groups and subgroups

A group is a set with an operation that satisfies closure, associativity, identity, and inverses. A subgroup is a subset that is itself a group under the same operation. When a subgroup is chosen, it can be shifted through the parent group in a systematic way to produce cosets.

1.2 Left cosets

Given a subgroup H of a group G and an element g in G, the left coset of H determined by g is the set gH = {gh : h in H}. This construction uses multiplication on the left by the fixed element g. Each left coset has the same size as H, though it may be a different subset unless g belongs to H.

1.3 Right cosets

The right coset of H determined by g is Hg = {hg : h in H}. Here the fixed element is placed on the right of each subgroup element. In commutative groups, left and right cosets are the same, but in noncommutative groups they can differ.

1.4 Notation and examples

Cosets are commonly written in the form gH for left cosets and Hg for right cosets. In the additive notation used for abelian groups, a coset of a subgroup H by an element a is often written a + H. For example, in the integers under addition, the subgroup of even integers has exactly two cosets: the even integers and the odd integers.

2 Properties of cosets

Cosets have several key structural properties that make them useful in abstract algebra. They divide a group into nonoverlapping pieces, each piece matching the size of the subgroup used to form it.

2.1 Partition of a group

The set of all left cosets of a subgroup H in a group G forms a partition of G. Every element of G lies in exactly one left coset of H. The same is true for right cosets. This means cosets organize the whole group into a family of equal-sized blocks.

2.2 Equal cardinality of cosets

Every coset of a subgroup has the same cardinality as the subgroup itself. The map h -> gh gives a one-to-one correspondence between H and gH, and similarly h -> hg gives a correspondence between H and Hg. This fact holds for finite and infinite groups alike.

2.3 Disjointness or equality

Any two cosets of the same subgroup are either identical or disjoint. If two cosets share even one element, then they contain exactly the same elements. This follows from the subgroup properties and is one of the reasons cosets form a partition.

2.4 Coset representatives

A coset representative is an element chosen from a coset to stand for that entire subset. Since many different elements may determine the same coset, representatives provide a convenient way to list or classify cosets. A complete set of representatives contains one element from each coset.

3 Cosets in subgroup theory

Cosets are especially important in describing how a subgroup sits inside a group. They give numerical and structural information about the subgroup, including its index and its relationship to quotient groups.

3.1 Index of a subgroup

3.2 Lagrange’s theorem

Lagrange’s theorem states that for a finite group G and a subgroup H, the order of H divides the order of G. This is a direct consequence of the fact that G is partitioned into cosets, each with the same size as H. The theorem has many consequences, including restrictions on possible subgroup orders.

3.3 Normal subgroups

A subgroup N is normal if gN = Ng for every element g in G. Normal subgroups are exactly the subgroups for which left and right cosets coincide. This symmetry allows cosets to be combined in a well-defined way and makes quotient groups possible.

3.4 Quotient groups

When a subgroup N is normal, the set of its cosets can be given a group structure called a quotient group or factor group. The operation is defined by combining representatives of cosets in a manner that does not depend on the choice of representative. Quotient groups are a major tool for simplifying groups and studying homomorphisms.

4 Special types of cosets

Beyond ordinary left and right cosets, algebra uses related constructions that reflect more elaborate ways of combining subgroup elements with group elements.

4.1 Left and right cosets in nonabelian groups

In a nonabelian group, the order of multiplication matters, so left and right cosets may differ. This distinction is important in understanding internal asymmetry within the group. Many examples in permutation groups and matrix groups illustrate this behavior.

4.2 Double cosets

A double coset has the form HgK, where H and K are subgroups of a group G and g is an element of G. It combines left and right subgroup action around a middle element. Double cosets appear in representation theory, combinatorics, and the study of symmetry relations.

4.3 Additive cosets in abelian groups

In abelian groups, cosets are often written using addition rather than multiplication. For a subgroup H and an element a, the coset is a + H = {a + h : h in H}. Because the group is commutative, there is no distinction between left and right forms, and the notation is often simpler.

5 Applications

Cosets arise throughout algebra and related fields whenever a structure is divided into equivalent pieces. They are particularly useful in arithmetic, symmetry, and the study of maps between groups.

5.1 Modular arithmetic

In modular arithmetic, congruence classes are cosets of the subgroup nZ inside the integers Z. All integers with the same remainder modulo n belong to the same coset. This viewpoint gives a natural group-theoretic interpretation of arithmetic modulo n.

5.2 Symmetry groups

Cosets help describe symmetry groups by grouping transformations that differ by elements of a chosen subgroup. In geometry and algebra, this can clarify how a larger symmetry set is built from smaller ones. Cosets also provide a way to count distinct symmetry configurations.

5.3 Factor groups and homomorphisms

Cosets are closely tied to homomorphisms, especially through kernels and quotient constructions. The kernel of a homomorphism is a normal subgroup, and the corresponding cosets form the factor group through which the map often factors. This relationship is a key part of the first isomorphism theorem.

5.4 Classification of equivalence classes

Cosets naturally define equivalence classes under the relation g ~ h if g^{-1}h lies in a subgroup H. This perspective classifies elements according to whether they differ by an element of H. It links group theory with the general theory of partitions and equivalence relations.

6 Related concepts

Cosets connect group theory to several broader ideas in algebra and discrete mathematics. These related notions help explain why cosets are so widely used.

6.1 Equivalence relations

An equivalence relation is a rule that divides a set into classes of mutually related elements. Cosets provide a standard example, since belonging to the same coset is an equivalence relation on a group. The resulting classes partition the group in a structured way.

6.2 Orbits under group actions

Orbits are sets of points reached by acting on an object with all elements of a group. Like cosets, orbits partition a set into disjoint classes. The analogy is especially strong because cosets can be viewed as orbits under the action of a subgroup on the parent group.

6.3 Subgroup lattices

The collection of all subgroups of a group can be arranged in a lattice by inclusion. Cosets help interpret how these subgroups sit inside one another by measuring index and comparing partitions. They therefore support a broader structural study of the subgroup lattice.