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Group Theory: Formation of a Group under an Associative Product

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Question: Let G be a nonempty set closed under an associative product, which in addition satisfies:
(a) There exists an e in G such that e.a = a for all a in G.
(b) Given a in G, there exists an element y(a) in G such that y(a).a = e.
Then G is a group under this product.

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Let be a nonempty set closed under an associative product, which in addition satisfies:
(a) There exists an such that for all
(b) Given , there exists an element such that .
Prove that must be a group under this product.

Solution: Let be a nonempty set closed under an associative product, which in addition satisfies:
(a) There exists an such that for all
(b) Given , there exists an element such that .

Since is closed under an associative product,
Closure Property holds in
and associative law also holds in .

Let be any ...

Solution provided by:
Thokchom Sarojkumar Sinha, MSc
Education
  • BSc, Manipur University
  • MSc, Kanpur University
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