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The following collection of cohomology functions is designed to provide
a flexible set of tools for computing with first and second cohomology
groups of any type of finite group acting on any reasonable module,
including a module defined by an action on an arbitrary finitely
generated abelian group. First (but not second)
cohomology groups can also be calculated for
infinite groups defined by a finite presentation.
Zero-cocycles, one-cocycles and two-cocycles may
be computed and identified. Extensions of modules by groups can be
constructed as finitely presented groups, or as PC-groups when the
acting group is a PC-group. It is also possible to compute a representative
set of extensions of the module by the group each of which is distinct up
to a group isomorphism fixing the module. These functions complement, but
do not completely supplant, an older collection of functions pertaining
to cohomology groups, Schur multiplicators and covering groups which
apply to permutation groups (see Chapter PERMUTATION GROUPS
on Permutation Groups).
The first cohomology group H1(G, M) is calculated as the nullspace of
a certain matrix. The details can be found in Section 5 of [CCH01].
This immediately allows manipulation and identification of one-cocycles.
The second cohomology group H2(G, M) is more difficult to compute.
While it can also be found as the nullspace of a suitable matrix, this
matrix can be uncomfortably large in big examples. For soluble groups
defined by a PC-presentation, the matrix corresponds to solving the
consistency equations for a PC-presentation of a general extension of
the module by the group, which depends on the number of group generators
rather than its order, and is manageable for quite large groups.
For permutation and matrix groups G, the size of the matrix for which
the nullspace is required is much larger, but can often be reduced to a
reasonable size by using a base and strong generating set for G.
In the case where only the dimension of H2(G, M) is required, and
M is a module over a finite field of prime order p, then the
calculation of this dimension can be reduced to the determination of
H2(Q, M) for a suitable collection of p-subgroups Q of G. The
latter calculation can be carried out efficiently using the PC-presentation
approach (see [Hol85b] for details).
To use the new functions, the user must initially invoke the function
CohomologyModule, which creates a special object for the group
action corresponding to the module, and all subsequent (new) cohomology
functions take this object as their first argument.
In the case of a finite permutation or matrix group G acting on a module
M over a prime field, the dimension of H2(G, M) may be found much more
quickly by executing CohomologicalDimension(CM, 2), where (CM) is the
cohomology module for the action of G on M, rather than by invoking
Dimension(CohomologyGroup(CM, 2)).
However, the former call does not allow the possibility of subsequent
calculations with two-cocycles or extensions.
The equivalent older function, CohomologicalDimension(G, M, 2);
(for a permutation group G) is often faster still for small examples,
but the new function will succeed on much larger examples than the old.
For the convenience of the reader, some of these older functions are
described in this section of the Handbook. For complete details about the
older functions, see the section on cohomology in the chapter on
Permutation Groups.
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