Abstract Algebra - Subgroup of Permutations

[iamalexalright]

Homework Statement

A is a subset of R and G is a set of permutations of A. Show that G is a subgroup of S_A (the group of all permutations of A). Write the table of G.


Onto the actual problem:
A is the set of all nonzero real numbers.
$G={e,f,g,h}$
where e is the identity element, f(x) = 1/x, g(x) = -x, h(x) = -1/x

Would this be the right way to do it?

For each combination of elements in G (call the elements a,b) I need to show
a*b is in G

I also need to show that the inverse of a is in G.

Here is where I get confused, I'll start with with a = e:

ee = e
ef = f
eg = g
eh = h

Okay, that is all good, now letting a = f:
fe = e
ff = e
fg = h
fh = g

now a = g:
ge = g !
gf = h
gg = g !
gh = e

now a = h:
he = h
hf = g !
hg = f
hh = g !

What am I doing wrong here?

 

[micromass]

The following are wrong:

fe=e
gg=g
gh=e
hg=f
hh=g

What did you do to obtain those?