The image contains the following mathematical statement:

"(e) Let $G$ be a group and $a, b \in G$. Prove: If $b$ is the inverse of $a$, then $b$ is the inverse of $a$."

To interpret this, it seems to be a question about proving the uniqueness of an inverse in a group.

Proof Outline

1. Definition of Inverse in a Group:

   • In a group $G$, each element has a unique inverse.
   • For any element $a \in G$, there exists an element $b \in G$ such that $a \cdot b = e$ and $b \cdot a = e$, where $e$ is the identity element of the group.

2. Uniqueness of the Inverse:

   • Suppose $b$ is an inverse of $a$ (i.e., $a \cdot b = e$ and $b \cdot a = e$).
   • Assume there exists another element $c \in G$ such that $a \cdot c = e$ and $c \cdot a = e$.
   • We want to show that $b = c$.

3. Derivation:

   • Since $a \cdot b = e$ and $a \cdot c = e$, we can multiply both sides of $a \cdot c = e$ by $b$ (from the right): $a \cdot c \cdot b = e \cdot b.$
   • Simplifying, we get $c = b$, proving that the inverse is unique.

Would you like further details or have any questions?

Related Questions

1. What is the definition of a group in abstract algebra?
2. Why is the identity element unique in a group?
3. How can we prove that the inverse of an inverse returns the original element in a group?
4. Can there be multiple identity elements in a group?
5. How is associativity crucial in proving the uniqueness of inverses?

Tip

In group theory, always ensure that you clearly identify the properties of the group (e.g., associativity, identity, and inverses) as these are foundational to all proofs in this area.