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Math Help - counterexample, function help

  1. #1
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    counterexample, function help

    Hey, I can't seem to get anywhere with this big question so if someone could work through this question with me so I understand that would be greatly appreciated.

    a) If g \in M_n give a counterexample to g(x+y)=g(x)+g(y)

    b) Prove g(\frac{1}{a}x_1 +...+ \frac{1}{a}x_a)=\frac{1}{a}g(x_1)+...+\frac{1}{a}g  (x_a)

    c) Using part b, prove that if G \le M_n is finite then G fixes at least one element k \in R^n

    I think for part c that if |G|=a, I can then pick some x \in R^n and consider x_i = h_ix for each element h_1,...,h_a \in G.
    But I can't get any of part a,b or c anyway.


    Thanks guys for any help you can give me.
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  2. #2
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    Re: counterexample, function help

    what is M_n?
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  3. #3
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    Re: counterexample, function help

    M is the group of all rigid motions of the plane, n is dimension.
    Last edited by Juneu436; October 14th 2011 at 05:03 AM.
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  4. #4
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    Re: counterexample, function help

    do these rigid motions have to be origin-preserving? if not, then we can let g(x) = x + a, where a is any (non-origin) point in R^n.

    then g(x+y) = x + y + a, whereas g(x) + g(y) = x + y + 2a.
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  5. #5
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    Re: counterexample, function help

    Quote Originally Posted by Deveno View Post
    do these rigid motions have to be origin-preserving? if not, then we can let g(x) = x + a, where a is any (non-origin) point in R^n.

    then g(x+y) = x + y + a, whereas g(x) + g(y) = x + y + 2a.
    well M is the coarsest classification of orientation-preserving and orientation-reversing motion, and g(x)=x+a is a translation and thus, orientation-preserving. So you answer makes perfect sense, thanks.

    Any thoughts on part b and c?
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  6. #6
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    Re: counterexample, function help

    I have thought of an approach for part c, but it does not use part b.

    Any help here.

    Thanks
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  7. #7
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    Re: counterexample, function help

    I have thought of doing it like [tex]g(\frac{1}{a}
    sorry gys my computer is stuffing up and my working is wrong, any help
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  8. #8
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    Re: counterexample, function help

    If G is finite then every element of G must be a rotation or a reflection. So I get the rough idea behind part c, but I can't get part b.

    Any help at all?
    Thanks
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