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Prove that K is a Group

Source: ILL 1970 - Problem 28.

May 24, 2011
functionsuperior algebrasuperior algebra unsolved

Problem Statement

A set GG with elements u,v,w...u,v,w... is a Group if the following conditions are fulfilled: (i)(\text{i}) There is a binary operation ∘\circ defined on GG such that ∀{u,v}∈G\forall \{u,v\}\in G there is a w∈Gw\in G with u∘v=wu\circ v = w. (ii)(\text{ii}) This operation is associative; i.e. (u∘v)∘w=u∘(v∘w)(u\circ v)\circ w = u\circ (v\circ w) ∀{u,v,w}∈G\forall\{u,v,w\}\in G. (iii)(\text{iii}) ∀{u,v}∈G\forall \{u,v\}\in G, there exists an element x∈Gx\in G such that u∘x=vu\circ x = v, and an element y∈Gy\in G such that y∘u=vy\circ u = v.
Let KK be a set of all real numbers greater than 11. On KK is defined an operation by a∘b=ab−(a2−1)(b2−1) a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}. Prove that KK is a Group.
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