Inmathematics, theorder of afinite group is the number of its elements. If agroup is not finite, one says that its order isinfinite. Theorder of an element of a group (also calledperiod length orperiod) is the order of the subgroup generated by the element. If the group operation is denoted as amultiplication, the order of an elementa of a group, is thus the smallestpositive integerm such thata^m =e, wheree denotes theidentity element of the group, anda^m denotes the product ofm copies ofa. If no suchm exists, the order ofa is infinite.

The order of a groupG is denoted byord(G) or|G|, and the order of an elementa is denoted byord(a) or|a|, instead of${\displaystyle \mathrm{ord}(⟨a⟩),}$ where the brackets denote the generated group.

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

Thesymmetric group S₃ has the followingmultiplication 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, soord(S₃) = 6. By definition, the order of the identity,e, is one, sincee¹ =e. Each ofs,t, andw squares toe, so these group elements have order two:|s| = |t| = |w| = 2. Finally,u andv have order 3, sinceu³ =vu =e, andv³ =uv =e.

The order of a groupG and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated thefactorization of |G|, the more complicated the structure ofG.

For |G| = 1, the group istrivial. In any group, only the identity elementa = e has ord(a) = 1. If every non-identity element inG is equal to its inverse (so thata² =e), then ord(a) = 2; this impliesG isabeliansince${\displaystyle ab=(ab{)}^{-1}=b^{-1}{a}^{-1}=ba}$ . The converse is not true; for example, the (additive)cyclic groupZ₆ of integersmodulo 6 is abelian, but the number 2 has order 3:

${\displaystyle 2+2+2=6\equiv 0(\mathrm{mod}6)}$ .

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

${\displaystyle ⟨a⟩=\{a^k:k\in \mathbb{Z}\}}$ 

for thesubgroupgenerated bya, then

${\displaystyle \mathrm{ord}(a)=\mathrm{ord}(⟨a⟩).}$ 

For any integerk, we have

a^k =e   if and only if   ord(a)dividesk.

In general, the order of any subgroup ofG divides the order ofG. More precisely: ifH is a subgroup ofG, then

ord(G) / ord(H) = [G :H], where [G :H] is called theindex ofH inG, an integer. This isLagrange'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 forfinite groups: ifd divides the order of a groupG andd is aprime number, then there exists an element of orderd inG (this is sometimes calledCauchy's theorem). The statement does not hold forcomposite orders, e.g. theKlein four-group does not have an element of order four. This can be shown byinductive proof.^[1] The consequences of the theorem include: the order of a groupG is a power of a primep if and only if ord(a) is some power ofp for everya inG.^[2]

Ifa has infinite order, then all non-zero powers ofa have infinite order as well. Ifa has finite order, we have the following formula for the order of the powers ofa:

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

for every integerk. In particular,a and its inversea⁻¹ have the same order.

In any group,

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

There is no general formula relating the order of a productab to the orders ofa andb. In fact, it is possible that botha andb have finite order whileab has infinite order, or that botha andb have infinite order whileab has finite order. An example of the former isa(x) = 2−x,b(x) = 1−x withab(x) =x−1 in the group${\displaystyle Sym(\mathbb{Z})}$ . An example of the latter isa(x) =x+1,b(x) =x−1 withab(x) =x. Ifab =ba, we can at least say that ord(ab) divideslcm(ord(a), ord(b)). As a consequence, one can prove that in a finite abelian group, ifm denotes the maximum of all the orders of the group's elements, then every element's order dividesm.

SupposeG is a finite group of ordern, andd is a divisor ofn. The number of orderd elements inG is a multiple of φ(d) (possibly zero), where φ isEuler's totient function, giving the number of positive integers no larger thand andcoprime 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 composited such asd = 6, since φ(6) = 2, and there are zero elements of order 6 in S₃.

Group homomorphisms tend to reduce the orders of elements: iff: G → H is a homomorphism, anda is an element ofG of finite order, then ord(f(a)) divides ord(a). Iff isinjective, 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 homomorphismh: S₃ → Z₅, because every number except zero inZ₅ has order 5, which does not divide the orders 1, 2, and 3 of elements in S₃.) A further consequence is thatconjugate elements have the same order.

An important result about orders is theclass equation; it relates the order of a finite groupG to the order of itscenter Z(G) and the sizes of its non-trivialconjugacy classes:

${\displaystyle |G|=|Z(G)|+\sum _i{d}_i}$ 

where thed_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 inG of the representatives of the non-trivial conjugacy classes. For example, the center of S₃ is just the trivial group with the single elemente, and the equation reads |S₃| = 1+2+3.

• Torsion subgroup

• 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