For any subset ¬ of the affine plane R^2, let G(¬) denote the group of all affine transformations f of R2 such that f(¬) = ¬.
The elements of the group G = G(¬), where ¬ is the hyperbola xy = 1
in R^2; i think are: (ax|dy) & (by|cx) where ad=bc=1
Show that the elements of G preserving each of the two asymptotes of ¬ form a normal subgroup N of index 2 in G, isomorphic to the multiplicative group R* of non-zero real numbers, which acts regularly on ¬.
Do all the elements of G then preserve the two asymptotes? Any hints how to show the rest please?!
No idea what i'm doin really i'm afraid, but here goes nothing:
The index of a subgroup N of G is the number of distinct left
(or right) cosets of N in G. Even if N is not a normal subgroup, the corre-
spondence xN -> Nx is a bijection between the set of left cosets and the
set of right cosets. If the index of N in G is equal to 2, then N is a normal subgroup of G:
Showing that Nx = xN for each x in G. If x in N there is nothing
to prove. Hence, let x is not in N. Since the index is two, the disjoint union of N and xN make up all of G. Analogously the disjoint union of N & Nx = G.
Since N does not equal Nx as well as N does not equal xN, it follows that Nx = xN.
Multiplying this with x^-1 gives that N^x = N and hence N is a normal
Thanks, any help would be much appreciated. x