Find All Elements of Affine Group G(¬) for Hyperbola xy = 1

[bishbashbosh]

Homework Statement

For any subset ¬ of the affine plane R^2, let G(¬) denote the group of all affine transformations f of R^2 such that f(¬) = ¬.

Exercise: Find all the elements of the group G = G(¬), where ¬ is the hyperbola xy = 1
in R^2.

The Attempt at a Solution

Hey! I'm afraid I’m completely lost with this, any hints please on where to start would be very much appreciated. Thanks x

 

[Billy Bob]

Welcome to PF, bishbashbosh.

Hey! I'm afraid I’m completely lost with this, any hints please on where to start would be very much appreciated.

Definitions. What is your definition of an affine transformation?

Hopefully you can put that in the form

x' = function of x and y
y' = function of x and y

Here (x',y') denotes the image of (x,y), not derivatives.

Now use the fact that xy=1 must imply x'y'=1.

 

[bishbashbosh]

Thank you Billy Bob, glad to be here!

OK

Definition:
an affine transformation f of Rⁿ is any transformation of the form
f _A,t : x -> Ax + t
where A is an invertible n×n matrix; detA = 0, and t is an element of Rⁿ.

so ??
x' = ax+by+t₁
y' = cx+dy+t₂

x'y'=acx²+(ad +bc)xy+bdy²+axt₂+byt₂+cxt₁+dyt₁+t₂t₁

...?

Sorry, what then are a,b,c,d please?

Thanks again x

 

[bishbashbosh]

If x'y'=xy=1

Comparing coefficients:

ac=0
bd=0
ad+bc=1

& as detA ≠ 0, ad-bc ≠ 0

Either a=0 & d=0 or b=0 & c=0

So elements of G:
( ax ) ,( by )
( dy ) ( cx )

Correct?!

Further please, let H be the subgroup of G preserving each of the two branches of ¬. Determine the index of H in G, whether H is normal in G, and whether H is abelian.

Definitions:

A subgroup H ≤ G of G is a subset H c_ G which is a group with respect to the same operation as G; equivalently, H is non-empty and is closed with respect to products and inverses.
We say that H is a normal subgroup if gH = Hg for all g in G; these cosets gH
then form a quotient group G/H with gH.g’H = (gg’)H.

Let H be a subgroup of the group G and let a in G. Then aH = {ah | h in H} is called the left coset of H in G determined by a.
Similarly, Ha = {ha | h in H} is called the right coset of H in G determined by a.

Let H ≤ G and let G be written as the disjoint union of the left cosets of H. Then the number of the left cosets in this decomposition is called the index of H in G, written |G : H|.

A group (G,*) is called abelian if the binary operation is commutative, i.e. if for all a, b in G, a * b = b * a.

Any ideas on this one please?

Thanks x

 

[Billy Bob]

Pretty good. For the original question, matrix A should have a=d=0 or b=c=0, but continue further to obtain bc=1 in the first case and ad=1 in the second. Did you mention t1=t2=0?

If you write down some examples, like A=[2 0 | 0 1/2], then you have nice geometric meaning; in this example x-coordinate is doubled and y-coordinate is halved.

Consider enough examples and you will easily that some of these preserve the branches of the hyperbola and others interchange them.

 

[bishbashbosh]

Sorry, couple of typo's in there. I think i understand the original question now, many thanks to you!

Ok so for 2nd, H= (ax|dy) for ad=1 ?!
Do all elements of G preserve each of the two asymptotes of ¬ ?

What extent will results apply to the groups G' = G(¬') preserving other hyperbolas ¬' not equal to ¬ in R2?

x

 

[Billy Bob]

I almost made the same mistake I think you are making. But matrix A=[a 0 | 0 d] (ad=1) does not necessarily carry each branch to itself; it might switch the two. And matrix A=[0 b | c 0] (bc=1) does not necessarily switch the two. It might just reflect across the line y=x.

Don't forget negative numbers!

For different hyperbolas, I guess rotate the coordinate axes first and/or dilate so that you have xy=1.

 

[bishbashbosh]

Ahhh Thank you once more!