Group extension

Central group extension

A group extension of by

is called central iff is contained within the centre , group whence is abelian. In what follows we write additively and and multiplicatively, and write for any ,

Second cohomology

Identifying with the [[Abelian groups as Z-modules|corresponding -module]] equipped with the trivial representation of (thus a -module) may consider the Group cohomology, where the 2-cochains are maps¹

and the 2-cocycles are 2-cochains such that

and the 2-coboundaries are 2-cochains such that

for some 1-cochain . Thus, in particular, -bilinear maps are 2-cocycles. The second cohomology group is then

Correspondence between 2-cocycles and central extensions

Given any -section of we have ; and defines a 2-cycle. Conversely let be a 2-cocycle. Then the set is a group under the following multiplication

with identity , and we have the above central extension where

and for the associated section we have . Note iff for all .

Proof

That follows from the the fact cosets of partition . Next we claim

defines a 2-cocycle. Note that , hence the formula is well-defined. Letting denote the inverse of , we have

as required, where we have used centrality of .

Now given a 2-cocycle we define the following multiplication on the set

which clearly constitutes a monoid since

and likewise on the right. The inverse is easily seen to be given by

Thus the given multiplication makes the set a group which we denote . Clearly we have the central extension

where and are given above. Letting , we find Noting that

now

as claimed.

This correspondence has the property

Central extensions are equivalent iff their 2-cocycles for some sections are cohomologous. Thus there is a bijection between and equivalence classes of extensions.

Proof

Consider the central extension

and let be -sections of , and consider the corresponding 2-cycles defined by

Then, taking into account the fact implies ,

so

thus different sections of give cohomologous 2-cocycles. It immediately follows that equivalent central extensions will give cohomologous 2-cocycles.

For the converse, it is sufficient to show that given a central extension with a section such that and a corresponding 2-cycle , the induced extension on is equivalent. We show that the following commutes

where

and is an isomorphism. Note that for every , for unique and , so is a well-defined bijection. Further, for any and

so is a group isomorphism, and

so the diagram commutes, as required.

Special cases

• Central extension of an abelian group
  • Cyclic central extension of a free abelian group
    • 2 central extension of a free abelian group
  • 2 central extension of an elementary abelian 2-group (includes extraspecial 2-groups)

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §5.1, p. 103 ↩