Elementary Group Theory - Notation

Notation

The group is often referred to as "the group " or more simply as "" Nevertheless, the operation "" is fundamental to the description of the group. is usually read as "the group under ". When we wish to assert that is a group (for example, when stating a theorem), we say that " is a group under ".

The group operation can be interpreted in a great many ways. The generic notation for the group operation, identity element, and inverse of are respectively. Because the group operation associates, parentheses have only one necessary use in group theory: to set the scope of the inverse operation.

Group theory may also be notated:

• Additively by replacing the generic notation by, with "+" being infix. Additive notation is typically used when numerical addition or a commutative operation other than multiplication interprets the group operation;
• Multiplicatively by replacing the generic notation by . Infix "*" is often replaced by simple concatenation, as in standard algebra. Multiplicative notation is typically used when numerical multiplication or a noncommutative operation interprets the group operation.

Other notations are of course possible.