Abelian Groups

Date: 05/15/2000 at 20:17:12
From: Ed Reed
Subject: Abelian Groups

How do I prove that the operation @, defined by a@b = a^ln(b), is an 
abelian group for the set of positive real numbers not equal to 1?

Date: 05/16/2000 at 11:34:58
From: Doctor Schwa
Subject: Re: Abelian Groups

A group has to:

1) Have an identity x so that a@x = x@a = a. What value of x will work 
   here?

2) Have an inverse, so that for each a there's some number y such that 
   a@y = y@a = the identity you found in the last part. Can you 
   express y in terms of a?

3) Be associative, so that (a@b)@c = a@(b@c) for all a, b and c. Can 
   you check that with algebra?

An abelian group ALSO has to:

4) Be commutative, so that a@b = b@a for all a and b. Can you check 
   that with algebra?

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/