This article needs additional citations for verification . (June 2009) (Learn how and when to remove this template message) 
Algebraic structure → Group theory Group theory 

Infinite dimensional Lie group

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called 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 × H is a group operation on H. This is usually denoted H ≤ G, read as "H is a subgroup of G".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).^{ [1] }^{ [2] }
If H is a subgroup of G, then G is sometimes called an overgroup of H.
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.
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].
Lagrange's theorem states that for a finite group G and a subgroup H,
where G and 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 G.^{ [4] }^{ [5] }
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. More generally, if p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.
Let G be the cyclic group Z_{8} whose elements are
and whose group operation is addition modulo eight. Its Cayley table is
+  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 two 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 topleft 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.
Every group has as many small subgroups as neutral elements on the main diagonal:
The trivial group and twoelement groups Z_{2}. These small subgroups are not counted in the following list.
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it also acts on everything that is built on the structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.
In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.
In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^{−1} ∈ N for all g ∈ G and n ∈ N. The usual notation for this relation is .
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after JosephLouis Lagrange.
In algebra, the kernel of a homomorphism is generally the inverse image of 0. An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix.
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = g^{–1}ag. This is an equivalence relation whose equivalence classes are called conjugacy classes.
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.
In mathematics, the free groupF_{S} over a given set S consists of all words that can be built from members of S, considering two words different unless their equality follows from the group axioms. The members of S are called generators of F_{S}, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to F_{S} for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in one and only one way as a product of finitely many elements of S and their inverses.
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup gives rise to the concept of the outer automorphism group.
In mathematics, specifically group theory, given an element g of a group G and a subgroup H of G,
In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as a combination of finitely many elements of the subset and their inverses.
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G. This can be understood as an example of the group action of G on the elements of G.
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory.
In group theory, a branch of mathematics, the order of a group is its cardinality, that is, the number of elements in its set. The order of an elementa of a group, sometimes also called the period length or period of a, is the smallest positive integer m such that a^{m} = e, where e denotes the identity element of the group, and a^{m} denotes the product of m copies of a. If no such m exists, a is said to have infinite order.
In mathematics, D_{3} (sometimes also denoted by D_{6}) is the dihedral group of degree 3, which is isomorphic to the symmetric group S_{3} of degree 3. It is also the smallest possible nonabelian group.
In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the grouptheoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.