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Math Help - Group axioms

  1. #1
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    Group axioms

    Please help.....

    For the following sets, with the binary operation given, determine whether or not it forms a group, by checking the group axioms.
    (R,◦),where x◦y=(xy)/√3
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  2. #2
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    Re: Group axioms

    What are the group axioms? First, you need an identity element: e\circ x = x\circ e = \dfrac{ex}{\sqrt{3}} = x. By cancellation, we find e = \sqrt{3}. Next, check that associativity holds:

    \begin{align*}x\circ (y\circ z) & = \dfrac{ x \tfrac{yz}{ \sqrt{3} } }{ \sqrt{3} } \\ & = \dfrac{ \tfrac{xy}{\sqrt{3}} z }{ \sqrt{3} } \\ & = (x\circ y)\circ z\end{align*}

    This follows from associativity and commutativity of real numbers.

    Next, check that each element has an inverse. Let x \in \mathbb{R}. You want to find y \in \mathbb{R} so that x\circ y = y\circ x = e = \sqrt{3}. So, x\circ y = \dfrac{xy}{\sqrt{3}} = \sqrt{3}. Solving for y, we find y = \dfrac{3}{x}. Hence, y exists so long as x \neq 0.

    So, no, it is not a group, but \left(\mathbb{R}\setminus \{0\},\circ \right) is a group.
    Last edited by SlipEternal; November 5th 2013 at 06:08 AM.
    Thanks from topsquark
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  3. #3
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    Re: Group axioms

    Much appreciated!!!!! I couldn't figure out the inverse.
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