The question asks to prove two things for a group $G$:

1. The identity element $e$ in $G$ is unique.
2. Every element in $G$ has a unique inverse.

Let's break down each part:

1. Uniqueness of the identity element:

We need to show that if there are two identity elements, say $e_1$ and $e_2$, they must be the same.

• By the definition of the identity element, for any element $a \in G$, we know that: $e_1 \cdot a = a \quad \text{and} \quad e_2 \cdot a = a$ Now, using the identity property on the left-hand side of the equation $e_1 \cdot a = a$, we get: e_1 \cdot a = e_2 \quad \text{(because e_2 \cdot a = a)} This implies $e_1 = e_2$, proving that the identity element is unique.

2. Uniqueness of inverses:

Let $a \in G$ be any element, and suppose it has two inverses, say $b_1$ and $b_2$. We need to show that $b_1 = b_2$.

• By the definition of an inverse, we have: $a \cdot b_1 = e \quad \text{and} \quad a \cdot b_2 = e$ Now, using $a \cdot b_1 = e$, we can multiply both sides of this equation on the left by $b_2$: $b_2 \cdot (a \cdot b_1) = b_2 \cdot e$ Since $b_2 \cdot a = e$, this simplifies to: $b_2 \cdot a \cdot b_1 = b_2 \cdot e \quad \Rightarrow \quad e \cdot b_1 = b_2$ Finally, we have: $b_1 = b_2$ Hence, the inverse of any element is unique.

These two parts prove that the identity element is unique, and every element in $G$ has a unique inverse.

Would you like further details on any part of this?

Here are some related questions to explore:

1. What properties must a set have to form a group?
2. How does the associativity property affect the structure of a group?
3. What is a group homomorphism, and how is it related to group operations?
4. What is the difference between a group and a ring?
5. Can a set with more than one identity element still be a group?

Tip: Understanding the uniqueness of the identity and inverses is fundamental to grasping the basic structure of a group in abstract algebra.