Question

If \((G,*)\) is a group such that \((a*b)^2 = (a*a)*(b*b)\) for all \(a,b \in G\) then \(G\) is

• A. abelian
• B. finite
• C. infinite
• D. None of these

Hint:

Use cancellation laws to simplify the given condition.

Answer 1

The Correct Option is A

To determine the nature of the group \( (G, *) \) given the property \( (a*b)^2 = (a*a)*(b*b) \) for all \( a, b \in G \), let's analyze the property step-by-step and determine what it implies about the group's characteristics:

1. Let's consider two elements \( a \) and \( b \) in the group \( G \). We are given that: (a*b)^2 = a^2 * b^2 .
2. This means: a * b * a * b = a * a * b * b .
3. The above equation can be rewritten by interchanging the position of the terms: a * b * a * b = a * a * b * b \Rightarrow a * (b * a) * b = (a * a) * (b * b) .
4. In standard notation, this equates to: (a*b) * (a*b) = a*a * b*b .
5. Now, rearrange for \( b \): a * b = b * a .
6. The above equation shows us that the operation \( * \) is commutative for any \( a, b \in G \). Hence, the group \( G \) is an abelian group by definition, as all pairs of elements commute.

Thus, the group \( G \) is abelian.

Conclusion: Based on the given property, \( G \) must be an abelian group. Therefore, the correct answer is the option: abelian.