Cyclic central extension of a free abelian group

2 central extension of a free abelian group

Let be a free abelian group of finite rank, and let

be a central extension with associated commutator map , where

Properties

Induced extraspecial 2-group

Now is an elementary abelian 2-group, and we have an induced -bilinear form

By Correspondence between quadratic forms and alternating bilinear forms at 2 we have a quadratic form

with pullback

We may then define the central subgroup

whence (Internal direct product) is the kernel of the projection , and

is a central extension with associated squaring map ,¹ group thus is an extraspecial 2-group.

Proof

proof

Liftings of

Using this notation, a map is an automorphism in such that iff

for the pullback of some quadratic form with polar form ,² and we have

Proof

proof

See also

• 2 central extension of an elementary abelian 2-group

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develop | en | sembr

Footnotes

1. 1988. Vertex operator algebras and the Monster, ¶5.3.4, p. 111 ↩

2. 1988. Vertex operator algebras and the Monster, ¶5.4.3–5.4.4, p. 113 ↩