Order (group theory)

This article is about order in group theory. For other uses in mathematics, see Order (mathematics). For other uses, see Order (disambiguation).

For groups with an ordering relation, see partially ordered group and totally ordered group.

Examples of transformations with different orders: 90° rotation with order 4, shearing with infinite order, and their compositions with order 3.

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 ${\displaystyle \operatorname {ord} (\langle a\rangle ),}$ 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(S₃) = 6. By definition, the order of the identity, e, is one, since e¹ = 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 u³ = vu = e, and v³ = 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 ${\displaystyle ab=(ab)^{-1}=b^{-1}a^{-1}=ba}$. 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:

${\displaystyle 2+2+2=6\equiv 0{\pmod {6}}}$.

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

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

for the subgroup generated by a, then

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

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,

${\displaystyle \operatorname {ord} (ab)=\operatorname {ord} (ba)}$

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 ${\displaystyle Sym(\mathbb {Z} )}$. 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:

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

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.

See also

• Torsion subgroup

Notes

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

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