From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 9 11:01:44 1997
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Date: Tue, 9 Sep 1997 00:20:27 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: maximal abelian quotients
dan asks
> Does every group have a
> unique maximal Abelian quotient?
yes. let G be a group. it's not difficult to show that
1) the commutator subgroup G' is normal,
2) the quotient group G / G' is abelian, and
3) if G --> A is a homomorphism to any abelian group A ,
then G' is in the kernel, so there is a unique homomorphism
G / G' --> A such that the original homomorphism is the composite
G --> G / G' --> A .
this last one is kind of technical, but in the special case where
A = G / N for some normal subgroup N , it says that if G / N is
abelian, then N contains the commutator subgroup. thus, G / G'
is the maximal abelian quotient of G .
the quotient G / G' is sometimes written G^ab (the "abelianization" of G).
as you might guess, this is an important construction in group theory,
and it's one of the reasons why commutator subgroups are important.
mike