# Definition:Group Product/Product Element

(Redirected from Definition: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: $b a^{-1} x a b^{-1} = b a$

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