subgroup

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In
group theory 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 ...
, 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 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 ...

''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 ...

Basic properties of subgroups

*A subset ''H'' of a group ''G'' is a subgroup of ''G''
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 ...
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''−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
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''''G'', and ''H'' is a subgroup of ''G'' with identity ''e''''H'', then ''e''''H'' = ''e''''G''. *The
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''''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 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 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 ...
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 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 ...
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 is isomorphic to Z, then ''a'' is said to have ''infinite order''. *The subgroups of any given group form a
complete lattice 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 th ...
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 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 ...
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 left
coset 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 an ...
''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''1 ~ ''a''2
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''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'', : $\left[ G : H \right] =$ 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.

Example: Subgroups of Z8

Let ''G'' be the cyclic group Z8 whose elements are :$G = \left\$ 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.

Example: Subgroups of S4

Let S4 be the symmetric group on 4 elements. Below are all the subgroups of S4, listed according to the number of elements, in decreasing order.

24 elements

The whole group S4 is a subgroup of S4, of order 24. Its Cayley table is

2 elements

Each element of order 2 in S4 generates a subgroup of order 2. There are 9 such elements: the $\binom = 6$ Cyclic permutation#Transpositions, transpositions (2-cycles) and the three elements (12)(34), (13)(24), (14)(23).

1 element

The trivial group, trivial subgroup is the unique subgroup of order 1 in S4.

Other 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.