Order of Groups and Elements in Group Theory

Explore the concept of "order" in group theory, differentiating between the order of a group (number of elements) and the order of an element (smallest power resulting in the identity). This guide explains these concepts with illustrative examples and demonstrates their significance in understanding group structure.

Order of Groups and Elements in Discrete Mathematics

Understanding "Order" in Group Theory

In group theory, the term "order" has two related meanings:

Order of a Group: The number of elements in the group (its cardinality). We denote this as |G| or ord(G).
Order of an Element: For an element 'a' in a group, its order is the smallest positive integer 'm' such that a^m = e, where 'e' is the identity element of the group. If no such 'm' exists (meaning a^m is never equal to e), the element has infinite order.

Example: The Symmetric Group S₃

The symmetric group S₃ consists of all possible ways to arrange three objects. It has six elements. Therefore, ord(S₃) = 6. Let's look at the order of its elements:

(The multiplication table for S₃ would be included here, along with explanations of the order of each element: the identity 'e' has order 1, some elements have order 2, and some have order 3.)

Order and Group Structure

The order of a group and its elements reveals important information about the group's structure. A group of order 1 is called the trivial group. An element x has order 1 if and only if x is the identity element. A group where every element is its own inverse has order 2 (and is abelian).

Relationships Between Orders

If an element 'a' generates a subgroup , then ord(a) = ord().

If ord(a) divides k, then a^k = e.

Lagrange's Theorem: The order of a group is divisible by the order of any of its subgroups. Specifically, ord(G) / ord(H) = [G:H], where [G:H] is the index of subgroup H in G.

Cauchy's Theorem: If a prime number 'p' divides the order of a group G, then G contains an element of order p. (Note: This doesn't hold for composite numbers.)

If 'a' has finite order, then ord(a^k) = ord(a) / gcd(ord(a), k).

ord(a) = ord(a^-1)

Counting Elements by Order

Let G be a finite group of order n, and let d be a divisor of n. The number of elements of order d in G is a multiple of φ(d), where φ is Euler's totient function (which counts the number of positive integers less than or equal to d that are relatively prime to d).

Homomorphisms and Orders

Group homomorphisms (structure-preserving maps between groups) affect the order of elements. If f: G → H is a homomorphism, and 'a' in G has finite order, then ord(a) is divisible by ord(f(a)). If f is injective (one-to-one), then ord(f(a)) = ord(a).

Class Equation

The class equation is a valuable result connecting the order of a finite group to the sizes of its conjugacy classes:

|G| = |Z(G)| + Σ_i d_i

Where |G| is the order of the group, |Z(G)| is the order of its center (elements that commute with all other elements), and d_i are the sizes of the non-trivial conjugacy classes (which are divisors of |G| greater than 1).

Conclusion

The concept of order in group theory provides critical insights into the structure and properties of groups, leading to powerful theorems like Lagrange's and Cauchy's theorems.