Re: Problems from Herstein's topics in algebra

From: Ken Pledger (Ken.Pledger_at_mcs.vuw.ac.nz)
Date: 06/21/04 

Date: Mon, 21 Jun 2004 12:51:02 +1200

In article <YLpBc.3242$%47.38@news01.roc.ny>,
 "Troubled" <mkajumap@hotmail.com> wrote:

> On page 32 Dr. Herstein goes on to say let G be the set of 2x2 matrices of
> the form (a b -b a) (the 1st row is the 1st 2 numbers and the 2nd row the
> last 2 numbers) with the operation being standard matrix multiplication. ...
> He then asks if the multiplication if G remind you of anything. Well, I
> don't see anything special (other than it being closed). Can someone please
> give me a leading hint?

      Look closely at the entries in a typical product

( a b) ( c d)
(-b a) (-d c). You _have_ seen those expressions before.

>
> #2) If G is a group of order 5 I need to show that it is abelian. I did this
> for a group order 3 and 4 but am having difficulty with this one. I know
> that it is cyclic (and hence abelian) by LaGrange's theorem but the problem
> came up before subgroups and Lagrange thm. I basically need to know what
> elements of G to start off with. Of course I can 1st prove Lagrange's thm
> and then everything will easily follow ....

      and that's certainly the neatest way to do it. However, a very
elementary proof is possible, although much trickier for order 5 than
for orders up to 4. This question came up in sci.math in March, and I
gave quite a careful explanation in a thread called "Proving
associativity". If you can't find that, post here again.

            Ken Pledger.