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If (G, ⋅) is a group such that (ab)-1 = a-1 b-1, ∀ a, b ∈ G, then G is a/an
- a)Commutative semi group
- b)Abelian group
- c)Non-abelian group
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If (G,⋅) is a group such that (ab)-1= a-1b-1,∀ a, b∈ ...
If (G, +) is a group such that for all a, b, c in G, (a + b) + c = a + (b + c), then (G, +) is said to be associative.
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If (G,⋅) is a group such that (ab)-1= a-1b-1,∀ a, b∈ ...
A group is said to be abelian if (a*b) = (b*a) ∀a,b ∈G
Since (G, ⋅) is a group
∴ (ab)-1 = (b-1a-1) (1)
Given: (ab)–1 = a–1b–1 (2)
From (1) and (2)
(a*b) = (b*a)
Therefore it is abelian
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