Definition:Group Product/Product Element

Definition

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

Let $a, b \in G$ such that $ = a \circ b$.

Then $g$ is known as the product of $a$ and $b$.

Also known as

The term product can be referred to as product element if it is important to distinguish between this and the group law, also called the group product.

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