In group theory, 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 on *H*. This is usually represented notationally by *H* ≤ *G*, read as "*H* is a subgroup of *G*". Group theory is that branch of mathematics concerned with the study of groups. ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
A **proper subgroup** of a group *G* is a subgroup *H* which is a proper subset of *G* (i.e. *H* ≠ *G*). The **trivial subgroup** of any group is the subgroup {*e*} consisting of just the identity element. If *H* is a subgroup of *G*, then *G* is sometimes called an *overgroup* of *H*. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
The same definitions apply more generally when *G* is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group *G* is sometimes denoted by the ordered pair (*G*,*), usually to emphasize the operation * when *G* carries multiple algebraic or other structures. In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ...
In the following, we follow the usual convention of dropping * and writing the product *a***b* as simply *ab*. ## Basic properties of subgroups
*H* is a subgroup of the group *G* if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever *a* and *b* are in *H*, then *ab* and *a*^{−1} are also in *H*. These two conditions can be combined into one equivalent condition: whenever *a* and *b* are in *H*, then *ab*^{−1} is also in *H*.) In the case that *H* is finite, then *H* is a subgroup if and only if *H* is closed under products. (In this case, every element *a* of *H* generates a finite cyclic subgroup of *H*, and the inverse of *a* is then *a*^{−1} = *a*^{n − 1}, where *n* is the order of *a*. - The above condition can be stated in terms of a homomorphism; that is,
*H* is a subgroup of a group *G* if and only if *H* is a subset of *G* and there is an inclusion homomorphism (i.e., i(*a*) = *a* for every *a*) from *H* to *G*. - The identity of a subgroup is the identity of the group: if
*G* is a group with identity *e*_{G}, and *H* is a subgroup of *G* with identity *e*_{H}, then *e*_{H} = *e*_{G}. - The inverse of an element in a subgroup is the inverse of the element in the group: if
*H* is a subgroup of a group *G*, and *a* and *b* are elements of *H* such that *ab* = *ba* = *e*_{H}, then *ab* = *ba* = *e*_{G}. - The intersection of subgroups
*A* and *B* is again a subgroup. The union of subgroups *A* and *B* is a subgroup if and only if either *A* or *B* contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. - If
*S* is a subset of *G*, then there exists a minimum subgroup containing *S*, which can be found by taking the intersection of all of subgroups containing *S*; it is denoted by <*S*> and is said to be the subgroup generated by *S*. An element of *G* is in <*S*> if and only if it is a finite product of elements of *S* and their inverses. - Every element
*a* of a group *G* generates the cyclic subgroup <*a*>. If <*a*> is isomorphic to **Z**/*n***Z** for some positive integer *n*, then *n* is the smallest positive integer for which *a*^{n} = *e*, and *n* is called the *order* of *a*. If <*a*> is isomorphic to **Z**, then *a* is said to have *infinite order*. - The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup
*generated by* the set-theoretic union of the subgroups, not the set-theoretic union itself.) If *e* is the identity of *G*, then the trivial group {*e*} is the minimum subgroup of *G*, while the maximum subgroup is the group *G* itself. This article does not cite any references or sources. ...
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...
In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...
The lattice of subgroups of the dihedral group Dih4, represented as groups of rotations and reflections of a plane figure. ...
In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...
In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
## Example Let *G* be the abelian group whose elements are In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
*G*={0,2,4,6,1,3,5,7} and whose group operation is addition modulo eight. Its Cayley table is Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ...
A Cayley table is a representation of a product defined on a set G. It is a group-theoretic generalization of an addition or a multiplication table. ...
+ | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 | 0 | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 | 2 | 2 | 4 | 6 | 0 | 3 | 5 | 7 | 1 | 4 | 4 | 6 | 0 | 2 | 5 | 7 | 1 | 3 | 6 | 6 | 0 | 2 | 4 | 7 | 1 | 3 | 5 | 1 | 1 | 3 | 5 | 7 | 2 | 4 | 6 | 0 | 3 | 3 | 5 | 7 | 1 | 4 | 6 | 0 | 2 | 5 | 5 | 7 | 1 | 3 | 6 | 0 | 2 | 4 | 7 | 7 | 1 | 3 | 5 | 0 | 2 | 4 | 6 | This group has a pair of nontrivial subgroups: *J*={0,4} and *H*={0,2,4,6}, where *J* is also a subgroup of *H*. The Cayley table for *H* is the top-left quadrant of the Cayley table for *G*. The group *G* is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...
## Cosets and Lagrange's theorem Given a subgroup *H* and some *a* in G, we define the **left coset** *aH* = {*ah* : *h* in *H*}. Because *a* is invertible, the map φ : *H* → *aH* given by φ(*h*) = *ah* is a bijection. Furthermore, every element of *G* is contained in precisely one left coset of *H*; the left cosets are the equivalence classes corresponding to the equivalence relation *a*_{1} ~ *a*_{2} if and only if *a*_{1}^{−1}*a*_{2} is in *H*. The number of left cosets of *H* is called the *index* of *H* in *G* and is denoted by [*G* : *H*]. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then 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...
A bijective function. ...
In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
This article does not cite any references or sources. ...
Lagrange's theorem states that for a finite group *G* and a subgroup *H*, Lagranges theorem, in the mathematics of group theory, 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. It is named after Joseph Lagrange. ...
where o(*G*) and o(*H*) denote the orders of *G* and *H*, respectively. In particular, the order of every subgroup of *G* (and the order of every element of *G*) must be a divisor of o(*G*). In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
**Right cosets** are defined analogously: *Ha* = {*ha* : *h* in *H*}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [*G* : *H*]. If *aH* = *Ha* for every *a* in *G*, then *H* is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. 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...
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