Order (group theory)

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 (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, 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.

The order of a group $G$ is denoted by $ord(G)$ or $| G |$ , and the order of an element $a$ is denoted by $ord(a)$ or $| a |$ , instead of $\operatorname {ord} (\langle a\rangle ),$ where the brackets denote the generated group.

Lagrange's theorem states that for any subgroup $H$ of a finite group $G$ , the order of the subgroup divides the order of the group; that is, $| H |$ is a divisor of $| G |$ . In particular, the order $| a |$ of any element is a divisor of $| G |$ .

Example

The symmetric group S₃ has the following multiplication table.

•	e	s	t	u	v	w
e	e	s	t	u	v	w
s	s	e	v	w	t	u
t	t	u	e	s	w	v
u	u	t	w	v	e	s
v	v	w	s	e	u	t
w	w	v	u	t	s	e

This group has six elements, so $ord(S 3) = 6$ . By definition, the order of the identity, $e$ , is one, since $e 1 = e$ . Each of $s$ , $t$ , and $w$ squares to $e$ , so these group elements have order two: $| s | = | t | = | w | = 2$ . Finally, $u$ and $v$ have order 3, since $u 3 = vu = e$ , and $v 3 = uv = e$ .

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 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 a² = e), then ord(a) = 2; this implies G is abelian since $ab=(ab)^{-1}=b^{-1}a^{-1}=ba$ . The converse is not true; for example, the (additive) cyclic group Z₆ of integers modulo 6 is abelian, but the number 2 has order 3:

2+2+2=6\equiv 0{\pmod {6}}

.

The relationship between the two concepts of order is the following: if we write

\langle a\rangle =\{a^{k}\colon k\in \mathbb {Z} \}

for the subgroup generated by a, then

\operatorname {ord} (a)=\operatorname {ord} (\langle a\rangle ).

For any integer k, we have

a^k = e if and only if ord(a) 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 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 groups: if d divides the order of a group G and d is a prime number, then there exists an element of order d in G (this is sometimes called Cauchy's theorem). The statement does not hold for composite orders, e.g. the Klein four-group does not have an element of order four. This can be shown by inductive proof.^[1] 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.^[2]

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(a^k) = ord(a) / gcd(ord(a), k)^[3]

for every integer k. In particular, a and its inverse a⁻¹ have the same order.

In any group,

\operatorname {ord} (ab)=\operatorname {ord} (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 $Sym(\mathbb {Z} )$ . 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(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, giving the number of positive integers no larger than d and coprime 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 homomorphisms 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, then ord(f(a)) = ord(a). This can often be used to prove that there are no homomorphisms or no injective homomorphisms, between two explicitly 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 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 Z(G) and the sizes of its non-trivial conjugacy classes:

|G|=|Z(G)|+\sum _{i}d_{i}\;

where the d_i 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.

Notes

^ Conrad, Keith. "Proof of Cauchy's Theorem" (PDF). Archived from the original (PDF) on 2018-11-23. Retrieved May 14, 2011.
^ Conrad, Keith. "Consequences of Cauchy's Theorem" (PDF). Archived from the original (PDF) on 2018-07-12. Retrieved May 14, 2011.
^ Dummit, David; Foote, Richard. Abstract Algebra, ISBN 978-0471433347, pp. 57

References

Dummit, David; Foote, Richard. Abstract Algebra, ISBN 978-0471433347, pp. 20, 54–59, 90
Artin, Michael. Algebra, ISBN 0-13-004763-5, pp. 46–47

[1] Conrad, Keith. "Proof of Cauchy's Theorem" (PDF). Archived from the original (PDF) on 2018-11-23. Retrieved May 14, 2011.

[2] Conrad, Keith. "Consequences of Cauchy's Theorem" (PDF). Archived from the original (PDF) on 2018-07-12. Retrieved May 14, 2011.

[3] Dummit, David; Foote, Richard. Abstract Algebra, ISBN 978-0471433347, pp. 57

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