2-central extension of an elementary abelian 2-group

Let $𝐸 = 2 𝑛$ an an elementary abelian 2-group ( $ℤ 2$ -vector space) of rank $𝑛$ and consider a central extension

1 \to ℤ + 2 e x p ↪ ˆ 𝐸 𝜋 ↠ 𝐸 \to 1

where $𝜋 𝑥 = ―― 𝑥$ Given a $𝖲 𝖾 𝗍$ -section $𝑠 (-) : 𝐸 ↪ ˆ 𝐸$ of $𝜋$ , the associated squaring map is then

𝑞 : 𝐸 \to ℤ 2 𝑎 \mapsto l n (𝑠 2 𝑎)

which is a quadratic form independent of $𝑠 (-)$ . The polar form of $𝑞$ is the associated commutator map $𝑐 0$ , and we have a bijection between (arbitrary) central extensions of the above form and quadratic forms on $𝐸$ . Furthermore, a group $ˆ 𝐸$ is an extraspecial 2-group iff it is a central extension of the above form for which $𝑞$ is nondegenerate.¹ #m/thm/group

Proof

Clearly equivalent extensions determine the same squaring map. Noting that $𝜋 (𝑠 2 𝑎) = 2 𝑎 = 0$ , it follows that $𝑞$ is well defined, and since for any $𝑝 \in ℤ 2$ ,

l n (𝑠 𝑎 e 𝑝 𝑠 𝑎 e 𝑝) = l n (𝑠 2 𝑎 e 2 𝑝) = l n (𝑠 2 𝑎) + 2 𝑝 = l n (𝑠 2 𝑎)

so $𝑞$ is independent of the section chosen. Now let $𝑠 𝑎 𝑠 𝑏 e 𝑝 = 𝑠 𝑎 + 𝑏$ . Then

𝑏 𝑞 (𝑎, 𝑏) = 𝑞 (𝑎 + 𝑏) - 𝑞 (𝑎) - 𝑞 (𝑏) = l n (𝑠 2 𝑎 + 𝑏) - l n (𝑠 2 𝑎) - l n (𝑠 2 𝑏) = l n (𝑠 𝑎 𝑠 𝑏 e 𝑝 𝑠 𝑎 𝑠 𝑏 e 𝑝 𝑠 - 2 𝑎 𝑠 - 2 𝑏) = l n (𝑠 𝑎 𝑠 𝑏 𝑠 - 2 𝑎 𝑠 𝑎 𝑠 𝑏 𝑠 - 2 𝑏) = l n [𝑠 𝑎, 𝑠 𝑏] = 𝑐 0

as claimed.

Let $𝑞 : 𝐸 \to ℤ 2$ be a quadratic form, ${𝑥 𝑖} 𝑛 𝑖 = 1$ be a $ℤ 2$ -basis of $𝐸$ , and define a unique bilinear map so that

𝜀 0 : 𝐸 \times 𝐸 \to ℤ 2 (𝑥, 𝑥) \mapsto 𝑞 (𝑥) (𝑥 𝑖, 𝑥 𝑗) \mapsto 0 𝑖 < 𝑗

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

1 \to ℤ + 2 ↪ ˆ 𝐸 ↠ 𝐸 \to 1

with the 2-cocycle $𝜀 0$ and thus the squaring map $𝑞$ .

Now for uniqueness, suppose

1 \to ℤ + 2 ↪ 𝐵 𝜑 ↠ 𝐸 \to 1

is a central extension with squaring map $𝑞$ . Then the associated bilinear map is the polar form $𝑏 𝑞$ . Defining a $𝖲 𝖾 𝗍$ -section $𝑠 (-)$ of $𝜑$ so that

𝑠 (-) : 𝑛 \sum 𝑘 = 1 𝑚 𝑘 𝑥 𝑘 = 𝑛 \prod 𝑘 = 1 𝑠 𝑚 𝑘 𝑥 𝑘

it is easily shown that $𝜀 0$ is the corresponding 2-cocyle and so $(𝐵, 𝜑)$ is equivalent.

Properties

Automorphisms

Letting

A u t (ˆ 𝐸; e) = {𝜑 \in A u t ˆ 𝐸 : 𝜑 e x p = e x p} A u t (𝐸; 𝑞) = {𝜓 \in A u t 𝐸 : 𝑞 𝜓 = 𝑞} A u t (𝐸; 𝑐 0) = {𝜓 \in A u t 𝐸 : 𝑐 0 (𝜓, 𝜓) = 𝑐 0}

it follows $A u t (𝐸; 𝑞) \leq A u t (𝐸; 𝑐 0)$ , and we have the group extension

1 \to 𝖠 𝖻 (𝐸, ℤ 2) * ↪ A u t (ˆ 𝐸; e) 𝜋 ↠ A u t (𝐸; 𝑐 0) \to 1

where for $𝜆 \in 𝖠 𝖻 (𝐴, ℤ 2) ≅ ℤ 𝑛 2 ≅ 𝐸$ , #m/thm/group

𝜆 * : ˆ 𝐴 \to ˆ 𝐴 𝑥 \mapsto 𝑥 e 𝜆 ¯ 𝑥

cf. the analogous result for free abelian groups.² Furthermore, if $ˆ 𝐸$ is extraspecial then $A u t ˆ 𝐸 = A u t (ˆ 𝐸; e)$ , and the inner automorphisms are given by

I n n ˆ 𝐸 = k e r 𝜋 = 𝖠 𝖻 (𝐸, ℤ 2) * ≅ 𝐸

where the isomorphism is natural, giving the short exact sequences

#state/tidy | #lang/en | #SemBr

1988. Vertex operator algebras and the Monster, §5.3, pp. 108–110 ↩
1988. Vertex operator algebras and the Monster, ¶5.4.5, p. 114 ↩

2-central extension of an elementary abelian 2-group

Properties

Automorphisms

Footnotes