The problem asks you to prove two things about a group $G$:

1. Uniqueness of the identity element: Prove that the identity element $e$ of a group $G$ is unique.
2. Uniqueness of the inverse of each element: Prove that every element in $G$ has a unique inverse.

Let’s proceed step by step.

Proof of the uniqueness of the identity element:

Let $e$ be an identity element in the group $G$. By definition, for all elements $a \in G$, we have: $e \cdot a = a \cdot e = a$ Now, assume there exists another element $e' \in G$ such that $e'$ is also an identity element, meaning: $e' \cdot a = a \cdot e' = a \quad \text{for all } a \in G.$ To show $e = e'$, consider the equation: $e \cdot e' = e'.$ Since $e'$ is an identity element, we also have: $e \cdot e' = e.$ Thus, $e = e'$, proving that the identity element is unique.

Proof of the uniqueness of the inverse element:

Let $a \in G$ be any element in the group, and let $b$ and $c$ be two elements in $G$ such that both satisfy: $a \cdot b = e \quad \text{and} \quad a \cdot c = e.$ We need to prove that $b = c$.

Start by considering the equation: $b = b \cdot e \quad \text{(since } e \text{ is the identity element)}.$ Now, substitute $e = a \cdot c$ into this equation: $b = b \cdot (a \cdot c).$ By associativity of group multiplication: $b = (b \cdot a) \cdot c.$ Since $b \cdot a = e$ (by the definition of $b$ as the inverse of $a$), this simplifies to: $b = e \cdot c = c.$ Thus, $b = c$, proving that the inverse of each element is unique.

Conclusion:

Both parts of the statement are now proven:

1. The identity element is unique.
2. Each element in the group has a unique inverse.

Would you like further clarifications on this proof, or do you have any other related questions?

Here are 5 related questions to expand the understanding:

1. What is the significance of the identity element in a group structure?
2. How does associativity play a role in proving the uniqueness of inverses?
3. Can you provide an example of a non-abelian group and explore its identity element and inverses?
4. What is the difference between a group and a monoid, especially concerning the identity element?
5. How does the concept of inverses extend to other algebraic structures like rings or fields?

Tip: Always remember to use associativity when manipulating equations in group theory. It ensures consistency when combining multiple elements and operations.