Question

Equation \(x * a = b\) has in group \((G, *)\)

• A. unique solution \(b * a^{-1}\)
• B. unique solution \(a^{-1} * b\)
• C. unique solution \(a^{-1} * b^{-1}\)
• D. many solutions

Hint:

In groups, equations have unique solutions.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In a group, multiply both sides on right by \(a^{-1}\).
Step 2: Detailed Explanation:
\(x * a = b\)
Multiply right by \(a^{-1}\): \(x * a * a^{-1} = b * a^{-1}\)
\(x * e = b * a^{-1} \rightarrow x = b * a^{-1}\)
So unique solution.
Step 3: Final Answer:
Unique solution \(b * a^{-1}\).