HOME

TheInfoList




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 their changes (cal ...
, the order of a
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
is the number of its elements. If 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 ...
is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...
, the order of an element of a group, is thus the smallest
positive integer 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 ...
such that , where denotes the
identity element 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 ...
of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a
divisor 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 ...

divisor
of . In particular, the order of any element is a divisor of .


Example

The
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
S3 has the following
multiplication table 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 ...
. : This group has six elements, so . By definition, the order of the identity, , is one, since . Each of , , and squares to , so these group elements have order two: . Finally, and have order 3, since , and .


Order and structure

The order of a group ''G'' and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the
factorization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

factorization
of , ''G'', , the more complicated the structure of ''G''. For , ''G'', = 1, the group is trivial. In any group, only the identity element ''a = e'' has ord(''a)'' = 1. If every non-identity element in ''G'' is equal to its inverse (so that ''a''2 = ''e''), then ord(''a'') = 2; this implies ''G'' is abelian since ab=(ab)^=b^a^=ba. The converse is not true; for example, the (additive)
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
Z6 of integers modulo 6 is abelian, but the number 2 has order 3: :2+2+2=6 \equiv 0 \pmod . The relationship between the two concepts of order is the following: if we write :\langle a \rangle = \ for the
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 ...
generated by ''a'', then :\operatorname (a) = \operatorname(\langle a \rangle). For any integer ''k'', we have :''ak'' = ''e''   if and only if   ord(''a'')
divides In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

divides
''k''. In general, the order of any subgroup of ''G'' divides the order of ''G''. More precisely: if ''H'' is a subgroup of ''G'', then :ord(''G'') / ord(''H'') = 'G'' : ''H'' where 'G'' : ''H''is called the
index Index may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastructure in the ''Halo'' series ...
of ''H'' in ''G'', an integer. This is Lagrange's theorem. (This is, however, only true when G has finite order. If ord(''G'') = ∞, the quotient ord(''G'') / ord(''H'') does not make sense.) As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord(S3) = 6, the possible orders of the elements are 1, 2, 3 or 6. The following partial converse is true for
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s: if ''d'' divides the order of a group ''G'' and ''d'' is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, then there exists an element of order ''d'' in ''G'' (this is sometimes called Cauchy's theorem). The statement does not hold for
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials * ...
orders, e.g. the
Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
does not have an element of order four). This can be shown by inductive proof. The consequences of the theorem include: the order of a group ''G'' is a power of a prime ''p'' if and only if ord(''a'') is some power of ''p'' for every ''a'' in ''G''. If ''a'' has infinite order, then all non-zero powers of ''a'' have infinite order as well. If ''a'' has finite order, we have the following formula for the order of the powers of ''a'': :ord(''ak'') = ord(''a'') /
gcd
gcd
(ord(''a''), ''k'')Dummit, David; Foote, Richard. ''Abstract Algebra'', , pp. 57 for every integer ''k''. In particular, ''a'' and its inverse ''a''−1 have the same order. In any group, : \operatorname(ab) = \operatorname(ba) There is no general formula relating the order of a product ''ab'' to the orders of ''a'' and ''b''. In fact, it is possible that both ''a'' and ''b'' have finite order while ''ab'' has infinite order, or that both ''a'' and ''b'' have infinite order while ''ab'' has finite order. An example of the former is ''a''(''x'') = 2−''x'', ''b''(''x'') = 1−''x'' with ''ab''(''x'') = ''x''−1 in the group Sym(\mathbb). An example of the latter is ''a''(''x'') = ''x''+1, ''b''(''x'') = ''x''−1 with ''ab''(''x'') = ''x''. If ''ab'' = ''ba'', we can at least say that ord(''ab'') divides
lcm
lcm
(ord(''a''), ord(''b'')). As a consequence, one can prove that in a finite abelian group, if ''m'' denotes the maximum of all the orders of the group's elements, then every element's order divides ''m''.


Counting by order of elements

Suppose ''G'' is a finite group of order ''n'', and ''d'' is a divisor of ''n''. The number of order-''d''-elements in ''G'' is a multiple of φ(''d'') (possibly zero), where φ is
Euler's totient function In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...
, giving the number of positive integers no larger than ''d'' and
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
to it. For example, in the case of S3, φ(3) = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 1, and is only of limited utility for composite ''d'' such as ''d''=6, since φ(6)=2, and there are zero elements of order 6 in S3.


In relation to homomorphisms

Group homomorphism 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 ...

Group homomorphism
s tend to reduce the orders of elements: if ''f'': ''G'' → ''H'' is a homomorphism, and ''a'' is an element of ''G'' of finite order, then ord(''f''(''a'')) divides ord(''a''). If ''f'' is
injective 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 ...
, then ord(''f''(''a'')) = ord(''a''). This can often be used to prove that there are no (injective) homomorphisms between two concretely given groups. (For example, there can be no nontrivial homomorphism ''h'': S3 → Z5, because every number except zero in Z5 has order 5, which does not divide the orders 1, 2, and 3 of elements in S3.) A further consequence is that
conjugate elements In mathematics, in particular field theory (mathematics), field theory, the conjugate elements of an algebraic element ''α'', over a field extension ''L''/''K'', are the roots of the minimal polynomial (field theory), minimal polynomial ''p'' ...
have the same order.


Class equation

An important result about orders is the class equation; it relates the order of a finite group ''G'' to the order of its
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
Z(''G'') and the sizes of its non-trivial
conjugacy class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
es: :, G, = , Z(G), + \sum_d_i\; where the ''di'' are the sizes of the non-trivial conjugacy classes; these are proper divisors of , ''G'', bigger than one, and they are also equal to the indices of the centralizers in ''G'' of the representatives of the non-trivial conjugacy classes. For example, the center of S3 is just the trivial group with the single element ''e'', and the equation reads , S3,  = 1+2+3.


See also

*
Torsion subgroupIn the theory of abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (math ...


Notes


References

* Dummit, David; Foote, Richard. Abstract Algebra, , pp. 20, 54–59, 90 * Artin, Michael. Algebra, , pp. 46–47 {{Authority control Group theory Algebraic properties of elements