Order (group theory)

From Academic Kids

In group theory, the term order is used in two closely related senses:

  • the order of a group is its cardinality, i.e. the number of its elements;
  • the order of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, we say that a has infinite order.

We denote the order of a group G by ord(G) or |G| and the order of an element a by ord(a) or |a|.

The relation between the two concepts is the following: if we write

<a> = {ak : k an integer}

for the subgroup generated by a, then

ord(a) = ord(<a>).

For any integer k, we have

ak = 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 the index of H in G, an integer. This is Lagrange's theorem.

As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. The converse is true for cyclic groups: if G is cyclic and d divides the order of G, then there exists an element of order d in G. If G is abelian and d divides the order of G, then there exists a subgroup of order d in G. This is not in general true for non-abelian groups.

If a has infinite order, then all 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(ord(a), k)

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

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. 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.

If G is a finite group of order n, and d is a divisor of n, then the number of elements in G of order d is a multiple of φ(d).

Group homomorphisms tend to reduce the orders of elements: if f : GH 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 (injective) homomorphisms between two concretely given groups. A further consequence is that conjugate elements have the same order.

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:

<math>|G| = |Z(G)| + \sum_{i}d_i\;<math>

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 certain non-trivial proper subgroups of G.

Several deep questions about the orders of groups and their elements are contained in the various Burnside problems; some of these questions are still open.

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