Homology in Group Theory
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359 Urs Stammbach Eidgen6ssische Technische Hochschule, ZLirich/Schweiz
Homology in Group Theory
Springer-Verlag Berlin. Heidelberg • New York 1973
AMS Subject Classifications (1970): 20J05 ISBN 3-540-06569-5 Springer-Verlag Berlin • Heidelberg • N e w York ISBN 0-387-06569-5 Springer-Verlag N e w Y o r k • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin . Heidelberg 1973. Library of Congress Catalog Card Number 73-19547. Printed in Germany. Offsetdruck : J ulius Beltz, Hemsbach/Bergstr.
INTRODUCTION The purpose braist group
of
may
these
learn
theory,
Notes
something
second,
methods
are
Chapter
I introduces
able
to a c h i e v e the
II w e h a v e
of groups.
Together
Chapters
as
far as
III,
applications extensions central
with
to b e
determined
being
that
tools
are
trary
variety
in all the
by
to
homological
some b a s i c
notions
facts
in g r o u p
about
the
to the
theory.
(co)homology
[43]
this will
(co)homology
theory
for t h e s e N o t e s .
the c o r e but
of this
volume.
not entirely
in a v a r i e t y ,
on central
We present
disjoint
theorems
extensions,
~
the
functors.
The group
[26],
V
areas:
o n the
localization
here
(a g r o u p
to
H2
it
. These
say
lower of n i l -
, H2
make
their
by Hopf
In a c e r t a i n
isomorphic
to)
in o r d e r
the
guide
about
the h i s t o r y
second
H2
appearances
, H2 • of t h e s e as
Eilenberg-MacLane
however
in 1 9 0 4
line
to an a r b i -
functors
first
was
homological
generalize
[46],
to s t u d y
a mild
group
sense
that
of t o p i c s
the main
functors
something
is w e l l - k n o w n
the m u l t i p l i c a t o r
the a u t h o r ,
(co)homology
in p a p e r s
[20].
of
the c h o i c e
of applications
, V
second
functors
for e x a m p l e
introduced
areas
functor
Eckmann
complete;
the p r e f e r e n c e
functors
the p l a c e
homology
in a n y w a y
four
It m a y be
group,
theory
groups.
largely
older;
alge-
field.
introduction
kernel
theorems
see w h a t
VI of Hilton-Stammbach
it is n e e d e d
abelian
the homological of h o m o l o g y
may
the basic
Chapter
form
own
to
in four d i f f e r e n t
We do n o t c l a i m
[25],
in his
complete
IV, V, V I
series,
potent
theorist
reader
with
First,
applications
assembled
as a r e a s o n a b l e
of groups
about
the g r o u p
In C h a p t e r
serve
is t w o f o l d .
they are much
Schur
integral
projective
[72],
[73]
homology representations
iv of a group. H2(G,A) [71
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