From: wcw@math.psu.edu (William C Waterhouse)
Subject: Re: Concerning Scrieber Refinements
Date: 15 Apr 1999 20:25:23 GMT
Newsgroups: sci.math
In article <7f04vr$p1a$1@news4.svr.pol.co.uk>,
"C Mahon" writes:
>...
> I have been P Cohn's Book on Algebra and have been puzzled this term.
>
> Near the beginning, about Lattices. Mr Cohn talks about a Scrieber
> Refinement,
>
> In particular in regards to the fact that Subgroups form a Modular Lattice.
>
> It appears obvious that with Join being Maximal Inclusion and Meet being
> Intersection that the Subgroups of a group are a Modular Lattice. However
> the text refers to Scrieber Refinement.
It sounds as though this is a misprint and the topic involved is
the Schreier refinement theorem. It was originally stated for groups
but generalized to modular lattices (by Ore, I think). The lattice
version is this:
Theorem. Let A_0, A_1, ... A_n be a (weakly) decreasing chain
of elements in a modular lattice, and let B_0, B_1, ..., B_m
be another such chain with A_0 = B_0 and A_n = B_m. Then
it is possible to refine both chains by inserting additional
elements
A_{i-1} = A_{i,0}, A_{i,1}, ..., A_{i,m} = A_i,
B_{j-1} = B_{j.0}, B_{j,1}, ... , B_{j,n} = B_j,
in such a way that the quotients A_{i,j-1}/A_{i,j} and
B_{j,i-1}/B_{j,i} are projective.
The proof starts by setting A_{i,j} = A_i u (A{i-1} n B_j)
and B_{j,i} = B_j u (B_{j+1} n A_i).
Details can be found for instance in Marshall Hall, Jr.,
The Theory of Groups, p. 125.
For groups the theorem applies to subgroup chains where each is
normal in the one before it; the "projective" here becomes
group isomorphism, and this is one way of proving the
Jordan-Hoelder Theorem.
William C. Waterhouse
Penn State