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Coset (group theory)

A coset is a translate of a subgroup inside a group. Left and right cosets partition a group, give the index, and lead to quotient groups when the subgroup is normal.

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In abstract algebra a group G together with a subgroup H gives rise to cosets: translates of H by elements of G. Cosets encode how H sits inside G and are a basic tool in group theory for counting, classification and the construction of quotient groups.

Definition

For an element g in G the left coset of H determined by g is the set gH = { g h : h in H }. The right coset determined by g is Hg = { h g : h in H }. Each is a subset of G with the same size as H, but left and right cosets need not coincide unless H has a special property described below.

Basic properties

  • Every left (or right) coset has cardinality equal to |H|; multiplication by a group element is a bijection on G.
  • Left cosets are either disjoint or identical: if g1H and g2H meet then g1H = g2H. The same holds for right cosets.
  • Cosets of H partition G. This gives the index [G : H], the number of distinct cosets (finite or infinite), a fundamental invariant of the embedding of H in G.
  • There is an equivalence relation: a ~ b if a^{-1}b is in H; the equivalence classes are precisely the left cosets.

Normal subgroups and quotients

If every left coset equals the corresponding right coset (gH = Hg for all g in G) then H is called normal in G. Only for normal subgroups does the set of cosets carry a natural group structure, the quotient group G/H, with multiplication (gH)(kH) = (gk)H well defined. Many constructions in algebra and topology rely on quotient groups.

Examples and notable facts

The integers Z and the subgroup nZ provide a familiar example: cosets are the residue classes modulo n and the quotient Z/nZ is the cyclic group of order n. In a finite group, Lagrange's theorem follows by counting cosets: |G| = |H| [G : H], so the order of a subgroup divides the order of the group. Not every subgroup is normal: for instance some subgroups of the symmetric group are not invariant under conjugation, so left and right cosets differ. However, any subgroup of index 2 is automatically normal because there are only two cosets.

Further notions include coset representatives or transversals (a choice of one element from each coset) and double cosets HxK when two (possibly different) subgroups are involved; these play roles in counting arguments and in representation theory. Cosets therefore bridge structural, enumerative and computational aspects of group theory.

Access without mathematical prerequisites

Groups are used in mathematics to generalize arithmetic with numbers. Accordingly, a group consists of a set of things (e.g., numbers, symbols, objects, motions) and a computational rule (a conjunction, represented in this article as ∗ *) that specifies how to deal with these things. This computational rule must satisfy certain rules, called group axioms, which are explained below.

A set is said to be a group if, for a set together with a conjunction of each two elements of that set, written here as a*bthe following requirements are fulfilled:

  1. The conjunction of two elements of the set in turn gives an element of the same set. (Completeness)
  2. For the linkage, the bracketing is irrelevant, that is, it holds {\displaystyle (a*b)*c=a*(b*c)}for all a,b,c. (Associative Law)
  3. There is an element e in the set that does nothing with respect to the link, that is, a ∗ *-neutral element: {\displaystyle a*e=e*a=a}for all a.
  4. For every element a there is, with respect to the link, an inverse element, that is, a ∗ {\displaystyle *-inverse element a^{*}. This has the property of ayielding the neutral element when concatenated with : {\displaystyle a^{*}*a=a*a^{*}=e}.

Note that if there are multiple links on the set, such as ∗ *and \circ , then there are multiple neutral and inverse elements, each matching the linkage. If it is clear from the context that only one particular linkage is meant, then one briefly speaks of the neutral element eand the inverse element a^{*}to awithout explicitly mentioning the linkage again.

  • If, in addition, the operands may be interchanged, i.e., if always a * b = b * aholds, then we have an abelian group, also called a commutative group. (Commutative Law)

Examples of abelian groups are

  • the integers \mathbb {Z} with addition + as the link and zero as the neutral element,
  • the rational numbers \mathbb {Q} without zero with multiplication \cdot as the link and one as the neutral element. The zero must be excluded here, since it has no inverse element: "1/0" is not defined.

The very general definition of groups makes it possible to take not only sets of numbers with corresponding operations as groups, but also other mathematical objects with suitable linkages that satisfy the above requirements. Such an example is the set of rotations and reflections (symmetry transformations) by which a regular n-corner is mapped onto itself, with the successive execution of the transformations as a linkage (Dieder group).

Definition of a group

→ Main article: Group (mathematics)

A group is a pair (G,*). Here Ga set and ∗ *a two-digit link with respect to G. That is, this *\colon G\times G\to G,(a,b)\mapsto a*bdescribes the mapping ∗ Moreover, the following axioms for the linkage must be satisfied for (G,*)called a group:

A group (G,*)is called abelian or commutative if in addition the following axiom is satisfied:

Otherwise, i.e., if there exist group elements a,b\in Gfor which a*b\neq b*a, the group is called (G,*)nonabelian.

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