Definition:Group Product/Group Law

Definition

Let $\struct {G, \circ}$ be a group.

The operation $\circ$ can be referred to as the group law.

Also known as

The term group law is often referred to as the group product, but this can easily be confused with the product (element).

Other terms that can be seen are:

group operation

product rule

Some sources rely on the language of arithmetic and call it multiplication.

However, this is not recommended as it can cause the reader's to be confused into assuming that the elements of $G$ are numbers, when this is not necessarily so.

Examples of Operations on Group Product

Example: $b x a^{-1} = a^{-1} b$

$b x a^{-1} = a^{-1} b$

Example: $a x a^{-1} = e$

$a x a^{-1} = e$

Example: $a x a^{-1} = a$

$a x a^{-1} = a$

Example: $a x b = c$

$a x b = c$

Example: $b a^{-1} x a b^{-1} = b a$

$b a^{-1} x a b^{-1} = b a$

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26$