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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 often denoted *H* ≤ *G*, read as "*H* is a subgroup of *G*".

- Subgroup tests
- Basic properties of subgroups
- Cosets and Lagrange's theorem
- Example: Subgroups of Z8
- Example: Subgroups of S4
- 24 elements
- 12 elements
- 8 elements
- 6 elements
- 4 elements
- 3 elements
- 2 elements
- 1 element
- Other examples
- See also
- Notes
- References

The **trivial subgroup** of any group is the subgroup {*e*} consisting of just the identity element.^{ [1] }

A **proper subgroup** of a group *G* is a subgroup *H* which is a proper subset of *G* (that is, *H* ≠ *G*). This is often 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*}).^{ [2] }^{ [3] }

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.

Suppose that *G* is a group, and *H* is a subset of *G*. For now, assume that the group operation of *G* is written multiplicatively, denoted by juxtaposition.

- Then
*H*is a subgroup of*G*if and only if*H*is nonempty and closed under products and inverses.*Closed under products*means that for every*a*and*b*in*H*, the product*ab*is in*H*.*Closed under inverses*means that for every*a*in*H*, the inverse*a*^{−1}is in*H*. These two conditions can be combined into one, that for every*a*and*b*in*H*, the element*ab*^{−1}is in*H*, but it is more natural and usually just as easy to test the two closure conditions separately.^{ [4] } - When
*H*is*finite*, the test can be simplified:*H*is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element*a*of*H*generates a finite cyclic subgroup of*H*, say of order*n*, and then the inverse of*a*is*a*^{n−1}.^{ [4] }

If the group operation is instead denoted by addition, then *closed under products* should be replaced by *closed under addition*, which is the condition that for every *a* and *b* in *H*, the sum *a*+*b* is in *H*, and *closed under inverses* should be edited to say that for every *a* in *H*, the inverse −*a* is in *H*.

- 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}. - If
*H*is a subgroup of*G*, then the inclusion map*H*→*G*sending each element*a*of*H*to itself is a homomorphism. - The intersection of subgroups
*A*and*B*of*G*is again a subgroup of*G*.^{ [5] }For example, the intersection of the*x*-axis and*y*-axis in**R**^{2}under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of*G*is a subgroup of*G*. - The union of subgroups
*A*and*B*is a subgroup if and only if*A*⊆*B*or*B*⊆*A*. A non-example: 2**Z**∪ 3**Z**is not a subgroup of**Z**, because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the x-axis and the y-axis in**R**^{2}is not a subgroup of**R**^{2}. - If
*S*is a subset of*G*, then there exists a smallest subgroup containing*S*, namely the intersection of all of subgroups containing*S*; it is denoted by ⟨*S*⟩ and is called 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, possibly repeated.^{ [6] } - Every element
*a*of a group*G*generates a cyclic subgroup ⟨*a*⟩. If ⟨*a*⟩ is isomorphic to**Z**/*n***Z**(the integers mod*n*) 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.

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*|.^{ [7] }^{ [8] }

**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 8. Its Cayley table is

+ | 0 | 4 | 2 | 6 | 1 | 5 | 3 | 7 |
---|---|---|---|---|---|---|---|---|

0 | 0 | 4 | 2 | 6 | 1 | 5 | 3 | 7 |

4 | 4 | 0 | 6 | 2 | 5 | 1 | 7 | 3 |

2 | 2 | 6 | 4 | 0 | 3 | 7 | 5 | 1 |

6 | 6 | 2 | 0 | 4 | 7 | 3 | 1 | 5 |

1 | 1 | 5 | 3 | 7 | 2 | 6 | 4 | 0 |

5 | 5 | 1 | 7 | 3 | 6 | 2 | 0 | 4 |

3 | 3 | 7 | 5 | 1 | 4 | 0 | 6 | 2 |

7 | 7 | 3 | 1 | 5 | 0 | 4 | 2 | 6 |

This group has two nontrivial subgroups: ■*J* = {0, 4} and ■*H* = {0, 4, 2, 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 Cayley table for *J* is the top-left quadrant of the Cayley table for *H*. The group *G* is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.^{ [9] }

S_{4} is the symmetric group whose elements correspond to the permutations of 4 elements.

Below are all its subgroups, ordered by cardinality.

Each group (except those of cardinality 1 and 2) is represented by its Cayley table.

Like each group, S_{4} is a subgroup of itself.

The alternating group contains only the even permutations.

It is one of the two nontrivial proper normal subgroups of S_{4}. (The other one is its Klein subgroup.)

Each permutation p of order 2 generates a subgroup {1, *p*}. These are the permutations that have only 2-cycles:

- There are the 6 transpositions with one 2-cycle. (green background)
- And 3 permutations with two 2-cycles. (white background, bold numbers)

The trivial subgroup is the unique subgroup of order 1.

- The even integers form a subgroup 2
**Z**of the integer ring**Z**: the sum of two even integers is even, and the negative of an even integer is even. - An ideal in a ring is a subgroup of the additive group of .
- A linear subspace of a vector space is a subgroup of the additive group of vectors.
- In an abelian group, the elements of finite order form a subgroup called the torsion subgroup.

- ↑ Gallian 2013, p. 61.
- ↑ Hungerford 1974, p. 32.
- ↑ Artin 2011, p. 43.
- 1 2 Kurzweil & Stellmacher 1998, p. 4.
- ↑ Jacobson 2009, p. 41.
- ↑ Ash 2002.
- ↑ See a didactic proof in this video.
- ↑ Dummit & Foote 2004, p. 90.
- ↑ Gallian 2013, p. 81.

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 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 will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, 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 non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element.

In mathematics, a **semigroup** is an algebraic structure consisting of a set together with an associative internal binary operation on it.

In abstract algebra, the **symmetric group** defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .

In the mathematical field of group theory, **Lagrange's theorem** is a theorem that states that for any finite group G, the order of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group , not only is an integer, but its value is the index , defined as the number of left cosets of in .

In mathematics, especially group theory, two elements and of a group are **conjugate** if there is an element in the group such that This is an equivalence relation whose equivalence classes are called **conjugacy classes**. In other words, each conjugacy class is closed under for all elements in the group.

In group theory, a branch of abstract algebra in pure mathematics, a **cyclic group** or **monogenous group** is a group, denoted C_{n}, 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 an integer power of *g* in multiplicative notation, or as an integer multiple of *g* in additive notation. This element *g* is called a *generator* of the group.

In mathematics, the **free group***F*_{S} over a given set *S* consists of all words that can be built from members of *S*, considering two words to be 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 exactly one way as a product of finitely many elements of *S* and their inverses.

In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called **cosets**. There are *left cosets* and *right cosets*. Cosets have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [*G* : *H*].

In abstract algebra, a **generating set of a group** is a subset of the group set 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 a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group whose elements are the permutations of the underlying set of G. Explicitly,

In mathematics, specifically group theory, the **index** of a subgroup *H* in a group *G* is the number of left cosets of *H* in *G*, or equivalently, the number of right cosets of *H* in *G*. The index is denoted or or . Because *G* is the disjoint union of the left cosets and because each left coset has the same size as *H*, the index is related to the orders of the two groups by the formula

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

In mathematics, a **Cayley graph**, also known as a **Cayley color graph**, **Cayley diagram**, **group diagram**, or **color group**, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem, and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing expander graphs.

In mathematics, the **order** of a finite group is the number of its elements. If a group is not finite, one says that its order is *infinite*. The *order* of an element of a group is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element a of a group, is thus 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, the order of *a* is infinite.

In group theory, a field of mathematics, a **double coset** is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let *G* be a group, and let *H* and *K* be subgroups. Let *H* act on *G* by left multiplication and let *K* act on *G* by right multiplication. For each *x* in *G*, the **( H, K)-double coset of x** is the set

In mathematics, **D _{3}** (sometimes alternatively denoted by

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 group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

- Jacobson, Nathan (2009),
*Basic algebra*, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1 . - Hungerford, Thomas (1974),
*Algebra*(1st ed.), Springer-Verlag, ISBN 9780387905181 . - Artin, Michael (2011),
*Algebra*(2nd ed.), Prentice Hall, ISBN 9780132413770 . - Dummit, David S.; Foote, Richard M. (2004).
*Abstract algebra*(3rd ed.). Hoboken, NJ: Wiley. ISBN 9780471452348. OCLC 248917264. - Gallian, Joseph A. (2013).
*Contemporary abstract algebra*(8th ed.). Boston, MA: Brooks/Cole Cengage Learning. ISBN 978-1-133-59970-8. OCLC 807255720. - Kurzweil, Hans; Stellmacher, Bernd (1998).
*Theorie der endlichen Gruppen*. Springer-Lehrbuch. doi:10.1007/978-3-642-58816-7. - Ash, Robert B. (2002).
*Abstract Algebra: The Basic Graduate Year*. Department of Mathematics University of Illinois.

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