Help with Paradoxical Groups: Vectors, Finite Groups, F3 & Z

[JSG31883]

Hi, I need some serious help in paradoxical groups!

1) Given vectors v1,v2 in R2 and w1,w2 in R2 (none lieing on a line thru the origin), show that you can find a unique C such that Cv1=w1 and Cv2=w2.

2) Show that a finite group is not very paradoxical.

3) Is F3 paradoxical? Is Z?


THANKS!

 

[matt grime]

would you care to define paradoxical?

1 is easy if you pick a basis. , though that is unnecessary, just define a map satisfying such and extend by linaerity to all of R^2 and note that two independent vectors in R^2 are a basis (you mean w1 w2 not lying on the same line, and v1 v2 not lying on the same line).

 

[JSG31883]

Def of Paradoxical:

G acts on X, E is subset of X.
E is G-paradoxical if there exists pairwise disjoin sets A1, ... , An, B1,..., Bm inside E and g1,...,gn, h1,...,hm inside G with E=(union)(Ai)=(union)(Bj).

If X is metric space and G acts by isometries, and we have A's, B's, g's, and h's as above, we have G-very paradoxical.

 

[matt grime]

Why didn't you say it was to help you do an assignment in a rush... my interest has dropped off, sorry. YOu might consider that your definition of paradoxical requres G to act on a set (your examples in the question don't) and you have not given a quantification of "not very" for paradoxical.