Orders of elements and groups

Orders of elements and groups

Definition of Order

• The order of an element in a group denotes the minimum number of times the element needs to be composed with itself (via the group operation) to reach the identity.
• The order of an element a in the group is denoted as ord(a).

Calculating Order of an Element

• If a is an element of a group and a^n=e (where e is the identity element and n is a positive integer), then n is the order of the element a.

Order of a Group

• The order of the group itself refers to the number of elements in the group. This is usually denoted by

  G

  when the group is named G.

Subgroups and Order

• Subgroups are subsets of a group that are themselves groups.
• The order of a subgroup must also divide the order of the group, this is known as Lagrange’s Theorem.

Importance of Order

• Understanding the order of elements and groups is fundamental to the understanding of group structure and dynamics.
• Operations and theorem in group theory often rely on the concept of order, making it an essential component of abstract algebra.