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.
Please give me a hint.
Thank you in advance