From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: learning group theory
Date: 28 Jan 1999 18:52:25 GMT
Newsgroups: sci.math
In article <36ACDAAF.2FA7@iae.nl>, Nico Benschop wrote:
>> The converse of Lagrange's theorem holds, in fact, for all finite
>> nilpotent groups. The groups that it holds for are a proper 'subset'
>> of solvable groups, however.
>
>Interesting. I know what a nilpotent semigroup is: the "opposite"
>of a group, namely a semigroup S with one idempotent '0', generated
>by each x in S, with 0 as final iteration. So x^i = x^{i+1} = 0 for
>some i>0 (depending on x). . . . But what is a nilpotent group ?
A group G is nilpotent if it has a central series (of finite length)
{1} = G_0 < G_1 < ... < G_n = G
in which each G_i/G_(i-1) lies in the center of G/G_(i-1).
Equivalently, all commutators in [G, G_i] must lie in G_(i-1).
For finite groups G, this turns out to be equivalent to being the
direct product of p-groups; since nearly every interesting property of
groups is easily reduced to direct factors, when they exist,
this means the study of finite nilpotent groups reduces to the study
of finite p-groups, about which quite a lot is known.
(Infinite groups are a bit trickier. Not all p-groups are nilpotent, and
I don't think there's such a nice structure theorem for infinite nilpotent
groups, unless you're willing to assume some tameness.)
I confess I don't really know why the name "nilpotent" is used since, I
agree, it doesn't quite match what is used in semigroups, ring theory, etc.
You can see the similarities if you focus on the commutator
[g,h]=(hg)^(-1)(gh) rather than the product gh in the group. Nilpotent
groups are those which in this way acquire a structure rather like a
Lie ring, and for Lie groups the connection is particularly tight.
Possibly Lie himself transferred the terminology from nilpotent Lie algebras
to nilpotent Lie groups, and then the corresponding group property was
axiomatized to the definition I gave above; but that's just speculation.
dave
==============================================================================
From: Joseph Rotman
Subject: nilpotent groups
Date: Thu, 28 Jan 1999 13:59:06 -0600 (CST)
Newsgroups: [missing]
To: rusin@vesuvius.math.niu.edu
You are almost right with why nilpotent groups
are so called. I don't have any etymological
authority, but I think the term was borrowed from
Lie algebras (by P. Hall?). But then the question
is why nilpotent Lie algebras are so called. But
this question is easy. Engel's theorem represents
such an algebra by a Lie algebra of matrices, and
each of these matrices is nilpotent in the sense
that some power is 0.
Joe Rotman
==============================================================================
From: Dave Rusin
Subject: Re: nilpotent groups
Date: Thu, 28 Jan 1999 14:01:12 -0600 (CST)
Newsgroups: [missing]
Thanks for the corroboration. Hall is a likely suspect since the terminology
was adopted when he was active (and few others were) -- certainly before 1940.
dave