Re: Finite groups exercise

On Tue, 07 Feb 2006 13:44:45 GMT, "Larry Hammick"
<larryhammick@xxxxxxxxx> wrote:

"David C. Ullrich"
Hint: This is a very easy problem :)

Yes, a question about algebra that _I_ see a solution to more
or less as soon as I start thinking about it is very easy.

Are you asking for a hint or wondering whether we can do it or what?
I was just being facetious :) To get this micro-theorem there's no need to
go into quotient groups or any such thing. Just assume that an odd homothety
(translation?) exists, and deduce in a flash that the odd homotheties are
equinumerous with the even, which is emphatically impossible, since the
total number of homotheties is odd.

Heh-heh. I don't see exactly why your obvious proof is right - why
are the odd translations (sorry, I know how to spell that)
equinumerous with the even ones?

My obvious argument: If P is one of those permutations and n is
the order of the group then P^n is the identity; otoh if P is
odd and n is odd then P^n is odd, hence not the identity.

But it came out of something not quite so trivial. I needed to characterize
finite commutative groups in which an odd homothety exists. And that, it
turn, came out of something really tough.
Hi again Dr. U, it has been a while.
Larry



************************

David C. Ullrich
.

Relevant Pages

  • Re: A possible argument for no more Fermat primes.
    ... substitution of variables in Algebra I??? ... What if n has an odd factor???? ... I am just showing the pattern of all 2^n+1 where ... Why are you trying to make this into a mystery? ...
    (sci.math)
  • Re: Surrogate factoring, theory versus implementation
    ... > not the underlying assumptions. ... > than competent at elementary algebra (as opposed to ... > think the disappointing results may be illustrating this ... so every product of two odd numbers is the difference of two squares. ...
    (sci.crypt)
  • Re: Surrogate factoring, theory versus implementation
    ... > not the underlying assumptions. ... > than competent at elementary algebra (as opposed to ... > think the disappointing results may be illustrating this ... so every product of two odd numbers is the difference of two squares. ...
    (sci.math)