# Group Theory: Formation of a Group under an Associative Product

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 Summary

This solution provides a response which shows the formation of a group under an associative product. The solution is detailed, well presented and has been completed in a Word document file which is attached.