need help with mathematical proof?

Question

show that the functions f=1/x, g=(x-1)/x generate a group of functions, the law of composition being composition of functions, which is isomorphic to the symmetric group S_3.

Answer 1

I agree with what Curt said. And the missing element is g^2, because it is the inverse of g..:) then mapping with S_3. Find the ones that has the same order, and map them.

Answer 2

Let's see. f^2 =1, obviously.

g = 1 - 1/x,. So g^2 = 1 - x/(x-1) = -1/(x-1). g^3 = g * g^2 = 1.

So we have an element of order 2 and an element of order 3. That's pretty encouraging.

It's easy to show that none of

1
f
g
fg
fg^2

equal each other.

Start a multiplication table with those 5 and it should become obvious what the 6th element is, and you'll be pretty much done.