Central extension of an abelian group

2 central extension of an elementary abelian 2-group

Let an an elementary abelian 2-group (-vector space) of rank and consider a central extension

where Given a -section of , the associated squaring map is then

which is a quadratic form independent of . The polar form of is the associated commutator map , and we have a bijection between (arbitrary) central extensions of the above form and quadratic forms on . Furthermore, a group is an extraspecial 2-group iff it is a central extension of the above form for which is nondegenerate.1 #m/thm/group

Proof

Clearly equivalent extensions determine the same squaring map. Noting that , it follows that is well defined, and since for any ,

so is independent of the section chosen. Now let . Then

as claimed.

Let be a quadratic form, be a -basis of , and define a unique bilinear map so that

Then by the Correspondence between 2-cocycles and central extensions there is a central extension

with the 2-cocycle and thus the squaring map .

Now for uniqueness, suppose

is a central extension with squaring map . Then the associated bilinear map is the polar form . Defining a -section of so that

it is easily shown that is the corresponding 2-cocyle and so is equivalent.

Properties

Automorphisms

Letting

it follows , and we have the group extension

where for , #m/thm/group

cf. the analogous result for free abelian groups.2 Furthermore, if is extraspecial then , and the inner automorphisms are given by

where the isomorphism is natural, giving the short exact sequences

https://q.uiver.app/#q=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Footnotes

  1. 1988. Vertex operator algebras and the Monster, §5.3, pp. 108–110 ↩

  2. 1988. Vertex operator algebras and the Monster, ¶5.4.5, p. 114 ↩