Let's consider the given scenario: the equation \(x * a = b\) in the group \((G, *)\). Our task is to determine whether there is a unique solution for \(x\), using the properties of groups.
- The group \((G, *)\) satisfies certain properties: closure, associativity, the presence of an identity element, and the existence of inverse elements for every element in the group.
- Given the equation \(x * a = b\), we want to isolate \(x\). To do so, we multiply both sides of the equation by the inverse of \(a\) (denoted as \(a^{-1}\)) on the right:
- Multiply both sides: \((x * a) * a^{-1} = b * a^{-1}\).
- Utilizing associativity, we can rewrite the left-hand side as:\(x * (a * a^{-1})\).
- Since \(a * a^{-1}\) equals the identity element \(e\) of the group, and multiplying any element by the identity does not change it, we simplify this to \(x * e = b * a^{-1}\).
- Thus \(x = b * a^{-1}\).
- Therefore, in the group \((G, *)\), the equation has the unique solution \(x = b * a^{-1}\).
By understanding the properties of groups and how inverse elements function, we can derive that the correct option is
- Unique solution: \(b * a^{-1}\)
Conclusion: The equation \(x * a = b\) in the group \((G, *)\) has a unique solution \(b * a^{-1}\).