# Composition of functions and stuff

1. Sep 4, 2009

### polarbears

1. Show that the set {f:R-{0,1}$$\rightarrow$$ R-{0,1}}, of functions under composition, is isomorphic to $$S _{3}$$
$$f_{1} = x$$
$$f_{2} = 1 - x$$
$$f_{3} = \frac {1}{x}$$
$$f_{4} = 1 - \frac {1}{x}$$
$$f_{5} = \frac {1}{1 - x}$$
$$f_{6} = \frac {x}{x - 1}$$

2. Relevant equations

3. The attempt at a solution

I don't really understand what the problem is asking

2. Sep 5, 2009

### lanedance

hey polarbears - there might be a smarter way, but I would start by having a look at the group S3, eg. all permutations of a set of 3 elements & see if you can find a 1-1 correspeondance between elements of S3 & the functions you are given, that is preserved under multiplication (in this case composition of functions)

for example, it should be clear that:
f1(fn(x))= fn(x), for any n, which makes it a good candidate for the identity element

info on S3 is here, have a look at the mult table in particular
http://groupprops.subwiki.org/wiki/Symmetric_group:S3

Last edited: Sep 5, 2009
3. Sep 5, 2009

### fmam3

Is this a question from the Gilbert book?