Order (group Theory) - Order and Structure

Order and Structure

The order of a group and that of an element tend to speak about the structure of the group. Roughly speaking, the more complicated the factorization of the order the more complicated the group.

If the order of group G is 1, then the group is called a trivial group. Given an element a, ord(a) = 1 if and only if a is the identity. If every (non-identity) element in G is the same as its inverse (so that a2 = e), then ord(a) = 2 and consequently G is abelian since by Elementary group theory. The converse of this statement is not true; for example, the (additive) cyclic group Z₆ of integers modulo 6 is abelian, but the number 2 has order 3:

.

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

for the subgroup generated by a, then

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) =, where 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. For example, in the symmetric group shown above, where ord(S₃) = 6, the orders of the elements are 1, 2, or 3.

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. 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 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.

In any group,

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 . An example of the latter is a(x) = x+1, b(x) = x-1 with ab(x) = id. 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.