If (G, .) is a group such that (ab)⁻¹ = a⁻¹ b⁻¹, for all a,b belongs to G,…
2016
If (G, .) is a group such that (ab)⁻¹ = a⁻¹ b⁻¹, for all a,b belongs to G, then G is a /an :
- A.
Commutative semi group
- B.
Abelian Group
- C.
Non-Abelian group
- D.
None of these
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Correct answer: B
In any group, (ab)^-1 = b^-1a^-1. If the group also has (ab)^-1 = a^-1b^-1 for all a and b, then a^-1b^-1 = b^-1a^-1, which implies ab = ba. Hence the group is abelian.
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