In mathematics, if G is a group, H a subgroup of G, and g an element of G, then Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation...
 gH = { gh : h an element of H } is a left coset of H in G, and
 Hg = { hg : h an element of H } is a right coset of H in G.
Some properties
We have gH = H if and only if g is an element of H. Any two left cosets are either identical or disjoint. The left cosets form a partition of G: every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and H itself is the only coset that is a subgroup. In mathematics, two sets are said to be disjoint if they have no element in common. ...
A partition of U into 6 blocks: a Venn diagram representation. ...
The left cosets of H in G are the equivalence classes under the equivalence relation on G given by x ~ y if and only if x^{ 1}y ∈ H. Similar statements are also true for right cosets. A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here. Some books on very applied group theory erroneously identify the conjugacy class as 'the' equivalence class as opposed to a particular type of equivalence class. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X  x ~ a } The notion of equivalence classes is useful for constructing sets out...
In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
Given a collection C of disjoint sets, a transversal is a set containing exactly one member of each of them. ...
All left cosets and all right cosets have the same number of elements (or cardinality in the case of an infinite H). Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H]. Lagrange's theorem allows us to compute the index in the case where G and H are finite, as per the formula: In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In mathematics, most commonly, Lagranges theorem states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. This can be shown using the concept of left cosets of H...
 G = [G : H] · H
This equation also holds in the case where the groups are infinite (but is somewhat less useful). The subgroup H is normal if and only if gH = Hg for all g in G. In this case one can turn the set of all cosets into a group, the factor group of G by H. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...
See Also Double coset In mathematics, an (H,K) double coset in G, where G is a group and H and K subgroups of G, is an equivalence class for the equivalence relation defined on G by x ~ y if there are h in H and k in K with hxk = y. ...
