Re: ATLAS


PaulHjelmstad wrote:
Well, I bought the ATLAS (withdrawn from a library,
only $80!) What a great book. I read the first 8
chapters right away, comprehending a fair amount,
with much being over my head, of course. My question
today is fairly
simple, could someone give me a simple definition
of the
Schur Multiplier? I have a rough idea, but am a
little
hung up on Central Extension. I have looked at
Wikipedia
(requires homology theory). Mathworld has nothing.

Derek Holt wrote:

I'll answer your first question!

The Schur Multiplier of a group G is the second
homolgy group H_2(G,Z)
(Z = integers).

OK, that probably wasn't very helpful, so here is the
equivalent group-
theoretical definition.

A central extension of G is defined to be a group
E with a
subgroup N <= Z(E) such that E/N ~= G,

or equivalently an exact sequence 1 -> N -> E -> G ->
1 with image(N)
<= Z(G).

Call it a "stem extension" if, in addition, N <=
[E,E], the commutator
subgroup of E.

We can order such extensions and say a central
extension (N',E') is
larger than (N,E) if (N,E) is a proper epimorphic
imageof (N',E') in
the sense that there is a commutative diagram

1 -> N' -> E' -> G -> 1

| | | 1
v v v

1 -> N -> E -> G -> 1

with all of the vertical maps epimorphic.

It can be proved that maximal stem extensions exist
for any group G.
Let (N,E) be one such. Then it turns out that N is
uniquely determined
up to isomorphism by G, and N is the Schur Multiplier
of G.

PH asks:

Thanks. I understand epimorphism, and commutator subgroups. I see that Z means integer, what exactly
does Z(E) and Z(G) mean? I am close to getting this,
thanks again for your explanation.

DH wrote:

In general, E is not uniquely determined by G. For
example, If G is a
Klein 4-group, then N is cyclic of order 2, but E can
be the dihedral
group D8 or the quaternion group Q8.

PH asks:

So N is 2, and E could be Dihedral(4 elements)?

DH wrote:

However, if G is perfect (and in particular if G is
simple), then E is
uniqely determined by G.

For example, if G = A5, then E = SL(2,5).

PH asks:

I see. What would N be in this case?

Thanks

PGH
.