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6. Let (G, *) be a group. Show that each equation of either the form ax = b or the form xa = b has a unique solution in G.

7. Show that (R - {1}, *), where a * b = a + b + ab is a group

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6. Let (G, *) be a group. Show that each equation of either the form ax = b or the form xa = b has a unique solution in G.

7. Show that (R - {1}, *), where a * b = a + b + ab is a group

[Recall:

Let G is a non-empty set, A binary operation * together with G defined as (G,*) is called a "Group" if the following properties hold in G:

G1: Closure property:

a * b  G for each a  G and b G

G2: Associative property:

a*(b*c) = (a*b)*c for each a G, b G and c G

G3: Existence of identity element:

a*e = e *a = a , e is the identity element in G.

G4: Existence of inverse element:

For each a G, there exists an ...

Solution Summary

This solution is comprised of a detailed explanation to show that each equation of either the form ax = b or the form xa = b has a unique solution in G.

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