Show that if G is a group such that (a * b)^2 = a^2 * b^2 for all a, b belonging to G, then G is abelian.

GATE 1988 · Discrete Mathematics · Group Theory · medium

Answer: a*b = b*a holds for all a, b in G, so G is abelian.

1. Expand both sides of the hypothesis: by associativity (a*b)^2 = (a*b)*(a*b) = a*b*a*b; the hypothesis sets this equal to a^2*b^2 = a*a*b*b, giving a*b*a*b = a*a*b*b
2. Cancel a on the left and b on the right: left-multiply by a^{-1} to cancel the leading a, getting b*a*b = a*b*b; then right-multiply by b^{-1} to cancel the trailing b, getting b*a = a*b for all a, b in G