In
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a branch of
mathematics, given a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''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
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
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, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
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 consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
, but this article will only deal with subgroups of groups.
Subgroup tests
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
''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.
*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.
If the group operation is instead denoted by addition, then ''closed under products'' should be replaced by the condition that for every ''a'' and ''b'' in ''H'', the sum ''a''+''b'' is in ''H'', and ''closed under inverses'' should be replaced by the condition that for every ''a'' in ''H'', the inverse −''a'' is in ''H''.
Basic properties of subgroups
*The
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
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
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
.
*The
intersection 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, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
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 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 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, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion.
In this lattice, the join of two subgroups is the subgroup generated by their uni ...
. (While the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
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 is the
minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
subgroup of ''G'', while the
maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
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, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
''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 and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
''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 .
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
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
of , ''G'', .
[See ]
didactic proof in this video
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
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 bina ...
Z
8 whose elements are
:
and whose group operation is
addition modulo 8. Its
Cayley table Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplic ...
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
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
Example: Subgroups of S4
Let S
4 be the
symmetric group
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 ...
on 4 elements.
Below are all the subgroups of S
4, listed according to the number of elements, in decreasing order.
24 elements
The whole group S
4 is a subgroup of S
4, of order 24. Its Cayley table is
12 elements
8 elements
6 elements
4 elements
3 elements
2 elements
Each element of order 2 in S
4 generates a subgroup of order 2.
There are 9 such elements: the
transpositions (2-cycles) and the three elements (12)(34), (13)(24), (14)(23).
1 element
The
trivial subgroup
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
is the unique subgroup of order 1 in S
4.
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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in a ring
is a subgroup of the additive group of
.
*A
linear subspace of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is a subgroup of the additive group of vectors.
*In an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, the elements of finite
order form a subgroup called the
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
.
See also
*
Cartan subgroup
In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate.
Examp ...
*
Fitting subgroup In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup ''F'' of a finite group ''G'', named after Hans Fitting, is the unique largest normal nilpotent subgroup of ''G''. Intuitively, it represents the small ...
*
Stable subgroup
*
Fixed-point subgroup
Notes
References
* .
* .
* .
*
*
*
* {{Cite book , last=Ash , first=Robert B. , url=https://faculty.math.illinois.edu/~r-ash/Algebra.html , title=Abstract Algebra: The Basic Graduate Year , date=2002 , publisher=Department of Mathematics University of Illinois , language=en
Group theory
Subgroup properties