Abelian Group Tables

Date: 04/29/99 at 06:56:49
From: Mel
Subject: How to construct abstract Abelian group tables?

Dr. Math,

Please could you help me with this.

How do I construct the first Abelian group for the general case?  I 
know it is of order 6. Could you give me some guidance about the 
associativity?

Thank you.

Date: 04/29/99 at 12:39:04
From: Doctor Rob
Subject: Re: How to construct abstract Abelian group tables?

Thanks for writing to Ask Dr. Math!

According to the Sylow Theorems, the group must contain elements of
orders 2 and 3. Let a^2 = e and b^3 = e, where e is the identity. Then 
look at a*b*a^(-1). It must also have order 3, but it can't equal b, 
because the group must be nonabelian. Thus it must equal b^2. So
a*b*a^(-1) = b^2. Now the six group elements are e, a, b, b*a, b^2, 
and b^2*a. You can multiply them together and use the relation
a*b = b^2*a to get the result back to one of these forms. For example,
if you want to multiply

   (b*a)*(b^2) = b*(a*b)*b,
               = b*(b^2*a)*b,
               = b^3*(a*b),
               = a*b,
               = b^2*a.

This should allow you to complete the required table.

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