From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: teaching kids the _wrong_ math
Date: 3 Oct 1999 07:47:53 GMT
Newsgroups: sci.math
Keywords: p-subgroups of GL(n,p)
In article <021019981358457075%bitbucket@home.com>,
Richard I. Pelletier wrote:
>The particular part of the Sylow Theorems that isn't obvious to me as a
>plausible _conjecture_ is "every p-subgroup is contained in some
>p-Sylow subgroup". I don't doubt its truth for a moment, but I want to
>see some evidence to motivate the conjecture in the first place.
>
>The only nontrivial examples (i.e. where there is a p-subgroup properly
>contained in a PSSG which is itself a proper subgroup) from which one
>might make that conjecture, in groups under order 16, are the 5 groups
>of order 12: yes, every subgroup of order 2 is contained in some
>subgroup of order 4. But to know _that_, I had to build and draw Hasse
>diagrams for all 5 groups of order 12.
>
>On the one hand, it may be reckless to make a conjecture about primes
>based on an observation about 2. On the other, it's good to make
>conjectures. But I want just a little more evidence.
Looking at groups of small order is just one way to collect data.
There are other good families of groups to look at. In this particular
case I would recommend the linear groups, e.g. GL(n,p). Compute the
order (hint: first row is nonzero, second is linearly independent, etc.),
compute the p-part, and compare to the order of U(n,p), the group of
upper-triangular matrices with 1's on the diagonal. So there's your
Sylow subgroup. Also you know that conjugacy amounts to changes of basis.
So now the proposition to be proved has a geometric
aspect: you must show that every p-group of linear maps fixes a maximal
flag (i.e. there is a sequence of subspaces 0 < V1 < V2 < ... < V_n
each of which is setwise fixed by the p-group, having dim(V_i) = i).
Prove that, and you have your family of examples to justify conjecturing
the same thing holds in any finite group.
dave
For finite group theory: http://www.math-atlas.org/index/20DXX.html
PS - you must've used non-ASCII characters in there. They don't work on
many screens (e.g., mine) so you might want to avoid their use.
[edited them --djr]