Re: Natural homomorphisms

jdm@xxxxxxxxxxxxxxx wrote:
> On page 3 of Elrifai and Morton's "Algorithms for positive braids",
> they state "The abelianisation map defines a natural homomorphism wt:
> B_n -> Z". But I can't understand how this can be a natural
> homomorphism when such a mapping is from some group G to G/N (N being
> a normal subgroup of G). Does anybody know what's going on here?

Well, I do not have the book and I do not know if the two above
instances of your notation match up i.e. if the Z above is a factor
group of B_n in a special particular way.

There might be a way of chosing for any given group G under consideration
a normal subgroup N_G of G in such a way that for a homomorphism
f: G ---> H you alwas have f(N_G) subset N_H.
(thing of the commutator subgroup for example)

Then the homomorphism G ---> H ---> H/N_H would indeed factor
uniquely through G ---> G/N_G such that the following diagram commutes

G ---> G/N_G

| |
| |
v v

H ---> H/N_H

this would make the passage from G to G/N_G a functor and
provide a natural transformation from the identity to that functor.

In fact, the term "abelianisation" above suggests, that the author
indeed uses the commutator subgroup.

Marc
.

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