Definition:Group Product
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Definition
Let $\struct {G, \circ}$ be a group.
The term group product can have two different interpretations:
Group Law
The operation $\circ$ can be referred to as the group law.
Product Element
Let $a, b \in G$ such that $ = a \circ b$.
Then $g$ is known as the product of $a$ and $b$.
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: $b a^{-1} x a b^{-1} = b a$
- $b a^{-1} x a b^{-1} = b a$