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

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

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

I think for part c that if $\displaystyle |G|=a$, I can then pick some $\displaystyle x \in R^n$ and consider $\displaystyle x_i = h_ix$ for each element $\displaystyle h_1,...,h_a \in G$.

But I can't get any of part a,b or c anyway.

Thanks guys for any help you can give me.

Re: counterexample, function help

what is $\displaystyle M_n$?

Re: counterexample, function help

M is the group of all rigid motions of the plane, n is dimension.

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.

Re: counterexample, function help

Quote:

Originally Posted by

**Deveno** 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?

Re: counterexample, function help

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

Any help here.

Thanks

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

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