In

^{−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.)
*When ''H'' is ''finite'', the test above 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}.)
*The _{''G''}, and ''H'' is a subgroup of ''G'' with identity ''e''_{''H''}, then ''e''_{''H''} = ''e''_{''G''}.
*The _{''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 ^{''n''} = ''e'', and ''n'' is called the ''order'' of ''a''. If is isomorphic to Z, then ''a'' is said to have ''infinite order''.
*The subgroups of any given group form a

_{1} ~ ''a''_{2} _{1}^{−1}''a''_{2} is in ''H''. The number of left cosets of ''H'' is called the index of a subgroup, index of ''H'' in ''G'' and is denoted by .
Lagrange's theorem (group theory), Lagrange's theorem states that for a finite group ''G'' and a subgroup ''H'',
: $[\; G\; :\; H\; ]\; =$
where , ''G'', and , ''H'', denote the order (group theory), 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'', .Dummit and Foote (2004), p. 90.
Right cosets are defined analogously: ''Ha'' = . They are also the equivalence classes for a suitable equivalence relation and their number is equal to .
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.

_{8} whose elements are
:$G\; =\; \backslash left\backslash $
and whose group operation is modular arithmetic, addition modulo 8. Its Cayley table is
This group has two nontrivial subgroups: and , 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 group, cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.

_{4} be the symmetric group on 4 elements.
Below are all the subgroups of S_{4}, listed according to the number of elements, in decreasing order.

_{4} is a subgroup of S_{4}, of order 24. Its Cayley table is

_{4} generates a subgroup of order 2.
There are 9 such elements: the $\backslash binom\; =\; 6$ Cyclic permutation#Transpositions, transpositions (2-cycles) and the three elements (12)(34), (13)(24), (14)(23).

_{4}.

group theory
In mathematics
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, a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, given a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

''G'' under a binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

∗, a subset
In mathematics
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''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 is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''".
The trivial subgroup of any group is the subgroup consisting of just the identity element.
A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ).
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
In mathematics, a semigroup is an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...

, but this article will only deal with subgroups of groups.
Basic properties of subgroups

*A subset ''H'' of a group ''G'' is a subgroup of ''G''if and only if
In logic
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it is nonempty and closed under products and inverses. (''Closure under products'' means that for every ''a'' and ''b'' in ''H'', the product ''ab'' is in ''H''. ''Closure under inverses'' means that for every ''a'' in ''H'', the inverse ''a''identity
Identity may refer to:
Social sciences
* Identity (social science)
Identity is the qualities, beliefs, personality, looks and/or expressions that make a person (self-identity
One's self-concept (also called self-construction, se ...

of a subgroup is the identity of the group: if ''G'' is a group with identity ''e''inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ad ...

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''homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

.
*The intersection
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of subgroups ''A'' and ''B'' of ''G'' is again a subgroup of ''G''. For example, the intersection of the ''x''-axis and ''y''-axis in R 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
Union commonly refers to:
* Trade union
A trade union (or a labor union in American English
American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of variety (linguistics), ...

of subgroups ''A'' and ''B'' is a subgroup if and only if ''A'' ⊆ ''B'' or ''B'' ⊆ ''A''. A non-example: 2Z ∪ 3Z 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 is not a subgroup of R.
*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 and is called the subgroup generated by ''S''. An element of ''G'' is in if and only if it is a finite product of elements of ''S'' and their inverses, possibly repeated.
*Every element ''a'' of a group ''G'' generates a cyclic subgroup . If is isomorphic
In mathematics
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to Z/''n''Z for some positive integer ''n'', then ''n'' is the smallest positive integer for which ''a''complete lattice
In mathematics
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under inclusion, called the lattice of subgroups
In mathematics, the lattice of subgroups of a Group (mathematics), group G is the Lattice (order), lattice whose elements are the subgroups of G, with the partial order Relation (mathematics), relation being set inclusion.
In this lattice, the join ...

. (While the infimum
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here is the usual set-theoretic intersection, the supremum
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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 is the minimum
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...

subgroup of ''G'', while the maximum
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...

subgroup is the group ''G'' itself.
Cosets and Lagrange's theorem

Given a subgroup ''H'' and some ''a'' in G, we define the leftcoset
In mathematics
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''aH'' = . 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''if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

''a'' Example: Subgroups of Z_{8}

Example: Subgroups of S_{4}

24 elements

The whole group S12 elements

8 elements

6 elements

4 elements

3 elements

2 elements

Each element of order 2 in S1 element

The trivial group, trivial subgroup is the unique subgroup of order 1 in SOther examples

*The even integers form a subgroup 2Z of the integer ring Z: the sum of two even integers is even, and the negative of an even integer is even. *An ideal (ring theory)#Definitions, ideal in a ring $R$ is a subgroup of the additive group of $R$. *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 (group theory), order form a subgroup called the torsion subgroup.See also

* Cartan subgroup * Fitting subgroup * Stable subgroup * Fixed-point subgroup * Subgroup testNotes

References

* . * . * . * {{Cite book, title=Abstract algebra, last1=Dummit, first1=David S., last2=Foote, first2=Richard M., date=2004, publisher=Wiley, isbn=9780471452348, edition=3rd, location=Hoboken, NJ, oclc=248917264 Group theory Subgroup properties