# Thread: finding a specific topological group with specific conditions

1. ## finding a specific topological group with specific conditions

Hi;

I have a question, it sounds difficult.

The question is the following:

Let X be a topological group such that the binary operation defined on it is *. For any two points a and b in X define a new operation by a(*)b=b^-1*a*b, [(*) is a new operation on X inherited from *]. By this (*), we get that a(*)a=a, for all a in X. Now let X_3 be a subset of X contains 3 elements, say a,b and c such that a(*)b=b(*)a=c; a(*)c=c(*)a=b; c(*)b=b(*)c=a. The question is find a specific example of topological group described above. Explain what is * and determine the three elements a,b and c

I thought about it and unfortunately I have not found any topological group with the conditions mentioned. We can define (*) for any group, yes, but the problem how to find the 3 elements in the question. I tried circle, torus, SO3. What do you think, can we find one as described in the question?

By topological group we mean a topological space and a group G at the same time such that the operation :G times G to G and taking the inverse: x to x^-1 are continuous functions. So it has algebraic structure (group) and topological structure. Examples: Circle, torus, SO3.

2. ## Re: finding a specific topological group with specific conditions

The operator (*) sounds like a change of basepoint operator (for homotopy classes), and the elements appear to be paths. In that case, the * operator would be concatenation of paths. My guesses at possible shapes: a genus 3 surface, a triangle, $\displaystyle \mathbb{R}^3$ where the operator is the cross-product and the elements are the x-, y-, and z- axes (using the full axes removes the +/- orientation), etc.

3. ## Re: finding a specific topological group with specific conditions

Thank you very much SlipEternal.

Also I tried to find one, which is SO3. Take any three transpositions of the symmetric group S3, for example (1 2), (2 3) and (1 2). Then if we conjugate any one by any other we get the third. SO3 is a topological group contains a copy of S3 . The three elements of S3 can be seen as rotations by 180 degrees around three axes lying in the plane, where each pair makes an angle of 120 degrees at the origin.

Thanks again SlipEternal.