Abelian GroupsDate: 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/ |
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