2 central extension of a free abelian group

Let $𝐴 = ℤ 𝑆$ be a free abelian group of finite rank, and let

1 \to 2 e x p ↪ ˆ 𝐴 𝜋 ↠ 𝐴 \to 1

be a central extension with associated commutator map $𝑐 0 : 𝐴 \times 𝐴 \to 2$ , where $𝜋 (𝑥) = ―― 𝑥$

Properties

Now $ˇ 𝐴 = 𝐴 / 2 𝐴$ is an elementary abelian 2-group, and we have an induced $ℤ 2$ -bilinear form

𝑐 1 : ˇ 𝐴 \times ˇ 𝐴 \to ℤ 2

𝑞 1 : ˇ 𝐴 \to ℤ 2

with pullback

𝑞 0 : 𝐴 \to ℤ 2

We may then define the central subgroup

𝐾 = {𝑥 2 e 𝑠 0 (―― 𝑥) : 𝑥 \in ˆ 𝐴}

whence $̂ 2 𝐴 = e ℤ 2 \times 𝐾$ (Internal direct product) is the kernel of the projection $ˆ 𝐴 \to ˇ 𝐴$ , and

1 \to 2 ↪ ˆ 𝐴 / 𝐾 \to ˇ 𝐴 \to 1

is a central extension with associated squaring map $𝑞 1$ ,¹ group thus $ˆ 𝐴 / 𝐾$ is an extraspecial 2-group.

Proof

proof

Using this notation, a map $𝜗 : ˆ 𝐴 \to ˆ 𝐴$ is an automorphism in $A u t (ˆ 𝐴; e)$ such that $𝜗 = - 1$ iff

𝜗 : 𝑥 \mapsto 𝑥 - 1 e 𝑞 0 (―― 𝑥)

for the pullback $𝑞 0$ of some quadratic form $𝑞 1$ with polar form $𝑐 0$ ,² and we have

𝐾 = {𝑥 2 e 𝑠 0 (―― 𝑥) : 𝑥 \in ˆ 𝐴} = {𝑥 𝜗 (𝑥 - 1) : 𝑥 \in ˆ 𝐴} = {𝑥 - 1 𝜗 (𝑥) : 𝑥 \in ˆ 𝐴} = {𝜗 (𝑥) 𝑥 - 1 : 𝑥 \in ˆ 𝐴}

Proof

proof