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Date: Tue, 9 Sep 1997 00:20:27 -0400
From: michael reid <reid@math.brown.edu>
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