Recent content by JSG31883

Let SL(2,Z) be the set of 2x2 matrices with integer coefficients. I know that SL(2,Z) is generated by S and T, where S= (0 -1 1 0) and T= (1 1 0 1). But how can I show that everyone element of G (the group generated by S and T) is in SL(2,Z)? Also, let...

How can I show that the group G=<a,b,c> with [a,b]=b, [a,c]=1, [b,c]=1 is solvable but not nilpotent? A group G is said to be nilpotent if G^i=identity for some i. A group G is said to be solvable if it has subnormal series G=GncG3cG2cG1=identity... where all quotient groups are abelian.

I am not really clear on what is meant by commutators. I know that the commutator of G is ABA^-1B^-1, but I am not sure how to check if a group is solvable by having the commutator eventually equal the trivial group. For example, I know that the Heisenberg group of 3x3 upper triangular...

Can someone help?

G acts via isometries on a set X, and A,B are subsets of X. Prove that the relation A~B is an equivalence relation on subsets of X iff A and B are G-equidecomposable. I think this has to do with the Banach-Schroder theorem, but am not sure. I know it is a definition in group theory, but...

3) No I don't know how to find the associated matrix... can you help?

1) SO(3) are all orthogonal matrices with det 1, and represents the rotation group in R3. I know what the rotation matrices in R3 look like, but don't know what the reflection ones do, so I can't say what happens to a point on the axis of reflection. 2) I do know that the composition of...

1) Let P,Q be planes through the origin in R3. Let Rp, Rq be the corresponding reflections. Is Rp*Rq (where * denotes "composition") in SO(3) or O(3)/SO(3)? What is the axis of rotation of Rp*Rq? 2) For a fixed A in SO(3) show that there are infinitely many pairs of planes P,Q such that...

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...

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...

for 2) how can I expand it out? You say if I expand it out I will be able to show it...

Can someone help me prove two theorems? I know they both are true, but can't come up with proofs. 1) Prove that a 3x3 matrix A is orientation preserving iff det(A)>0. 2) Prove that for A, B (both 3x3 matrices) that det(AB)=detA*detB. (A, B may or may not be invertible). THANK YOU!