Basic notions in group theory
category of groups
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

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 *. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two More precisely, H is a subgroup of G if the restriction of * to H is a group operation on H. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function This is usually represented notationally by H ≤ G, read as "H is a subgroup of G".

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.

The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation 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 the following, we follow the usual convention of dropping * and writing the product a*b as simply ab.

Contents 1 Basic properties of subgroups 2 Example 3 Cosets and Lagrange's theorem 4 See also

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. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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. In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the 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/nZ for some positive integer n, then n is the smallest positive integer for which aⁿ = e, and n is called the order of a. 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 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. In Mathematics, a complete lattice is a Partially ordered set in which all subsets have both a Supremum (join and an Infimum (meet In Mathematics, the lattice of subgroups of a group G is the lattice whose elements are the Subgroups of G with the (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. 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 elements of ) 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. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement

Example

Let G be the abelian group whose elements are

G={0,2,4,6,1,3,5,7}

and whose group operation is addition modulo eight. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers 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 a pair of nontrivial subgroups: J={0,4} and H={0,2,4,6}, where J is also a subgroup of H. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a 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 Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In general, subgroups of cyclic groups are also cyclic.

Cosets and Lagrange's theorem

Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH Because a is invertible, the map φ : H → aH given by φ(h) = ah is a bijection. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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 a₁ ~ a₂ if and only if a₁⁻¹a₂ is in H. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" ↔ 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. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In particular, the order of every subgroup of G (and the order of every element of G) must be a divisor of |G|. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without

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. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement.

See also

• Cartan subgroup
• Fitting subgroup
• stable subgroup

In Mathematics, a Cartan subgroup of a Lie group or Algebraic group G is one of the subgroups whose Lie algebra is a Cartan subalgebra In Mathematics, especially in the area of algebra known as Group theory, the Fitting subgroup F of a finite group In Abstract algebra, a branch of pure Mathematics, the Algebraic structure group with operators or Ω- group is a group with a

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