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Decide which of the following pairs $(G,*)$ are groups; for those which are, say what the identity and inverses are, and for the ones which aren't, say which axiom fails.

d. $G=(0,\infty)$ with $*$ given by $g*h :=\sqrt{gh}$

What does this $:=$ mean?

If it means what I think it means, which is that the binary operation is defined by $\sqrt{gh}$, then would the identity and inverse for $a\in G $ be $a$.

Also, how would I prove associativity i.e. $(g*h)*z=g*(h*z)$

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    $\begingroup$ Your interpretation is correct, but the Identity element must be the identity, independent of anything. $\endgroup$ – AlexR Mar 6 '15 at 1:36
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The notation $:=$ means "is defined to be". The set you give is not a group under that operation as there is no unique identity element.

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