From: Oliver Jenkinson 
Date: Mon, 5 Aug 1996 20:26:43 +0100
Subject: An elementary problem?


Dear colleagues,

The following question has been vexing me for several weeks now.
I've asked several algebraists about it, but as yet don't have a
definitive answer, though it appears to be an elementary problem.
If anyone could help in any way then I'd be much obliged.



Elementary linear algebra tells us that if two vector spaces have the
same dimension, and one contains the other, then in fact they are equal.

How far does this result generalise if we replace the underlying
field structure with just a group structure?

More precisely, suppose G is a group, and consider the direct product
(or sum) G^n.

Now suppose H is an `m-parameter subset' of G^n.
That is, the first m coordinates each vary freely over G, and the remaining
n-m coordinates depend on the first m coordinates.
So H is of the form

H={(g_1,..,g_m, w_{m+1}(g_1,...,g_m), .. ,w_n(g_1,...,g_m) ):g_1,..,g_m\in G}

where the w_i(g_1,...g_m) are words in g_1,...,g_m.

( Of course if G is abelian, then H is actually a subgroup of G^n, and the
words w_i(g_1,...,g_m) are Z-linear combinations of g_1,...g_m . )

Now suppose that H' is another `m-parameter subset' of G^n, though the m freely
varying coordinates need not be the first m. Again the remaining
coordinates are just words in the freely varying ones.

Question: If H is contained in H' then is it true that H=H' ?

Ideally I'd like to know which groups G give an affirmative answer to this
question. I'm hoping (perhaps naively) that it's true for _all_ groups.

If G is finite then it's certainly true, since |H|=|H'|=|G^m|.

If G is the integers then I'm fairly sure it's true as well, though my
proof is a bit sketchy.

Any help on this matter would be much appreciated - proofs, counterexamples,
references, or even just the correct language to couch this problem in.

 Yours,
  Oliver Jenkinson,
   omj@maths.warwick.ac.uk

--
8.8.96
Clarification:

Apologies for the ambiguity - yes, I did mean that the non-freely varying
coordinates should be 'G-words' in the freely varying ones, rather than
just having _some_ functional dependence.
I wanted them to be generalisations of linear (rather than non-linear)
vector subspaces.

 Oliver Jenkinson,