Abelian Group TablesDate: 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/ |
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