Definition:Group Product/Product Element

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

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