# Definition:Group Product/Group Law

< Definition:Group Product(Redirected from Definition:Group Operation)

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## 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: $\S 26$