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

Step 1: Formula / Definition}
\[ (a*b)^2 = a^2 * b^2 \]
Step 2: Calculation / Simplification}
\((a*b)*(a*b) = a*a*b*b\)
Left multiply by \(a^{-1}\): \(b*a*b = a*b*b\)
Right multiply by \(b^{-1}\): \(b*a = a*b\)
\(\therefore G\) is abelian.
Step 3: Final Answer
\[ \text{abelian} \]