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A product formula for the number of square roots of the identity in a central product of extraspecial $2$ -groups

Statement

Let $P_{1}$ and $P_{2}$ be extraspecial $2$ -groups and let $P = P_{1} \circ_{α} P_{2}$ be the central product along an isomorphism $α$ of their centres. Write $t (G) = ∣ {g \in G : g^{2} = 1} ∣$ . Then

$t (P) = \frac{t (P_{1}) t (P_{2}) + (∣ P_{1} ∣ - t (P_{1})) (∣ P_{2} ∣ - t (P_{2}))}{2} .$

Facts & Assumptions

Given: Extraspecial $2$ -groups $P_{1}, P_{2}$ with $Z (P_{i}) = ⟨ z_{i} ⟩$ of order two, an isomorphism $α : Z (P_{1}) \to Z (P_{2})$ , and $P = P_{1} \circ_{α} P_{2}$ with quotient map $π : P_{1} \times P_{2} \to P$ .

[F1]

For groups $G, H$ with central subgroups $Z_{1} \leq Z (G)$ , $Z_{2} \leq Z (H)$ and an isomorphism $α : Z_{1} \to Z_{2}$ , the central product $G \circ_{α} H$ is the quotient of $G \times H$ by $N = {(z, α (z)^{- 1}) : z \in Z_{1}}$ (The central product $G \circ_{α} H$ of two groups along an isomorphism of central subgroups).

[F2]

The external direct product $G \times H$ carries the componentwise operation (The external direct product $G \times H$ with componentwise multiplication).

[F3]

The quotient group $G / N$ has the left cosets $g N$ as elements, with product $(g N) (h N) : = g h N$ (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

[F4]

$Z (G) : = {z \in G : z g = g z for every g \in G}$ (The center $Z (G)$ of a group).

[L1]

For a finite $p$ -group $P$ the following are equivalent: $P$ is extraspecial; $P$ is nonabelian, $∣ Z (P) ∣ = p$ and $P / Z (P)$ is elementary abelian; $P$ is nonabelian and $Z (P) = P^{'} = Φ (P)$ has order $p$ (Three equivalent descriptions of an extraspecial $p$ -group).

[L2]

An elementary abelian $p$ -group is a finite abelian $p$ -group in which every nonidentity element has order $p$ (Elementary abelian $p$ -groups).

[L3]

The subgroup $N = {(z, α (z)^{- 1}) : z \in Z_{1}}$ of $G \times H$ is central, hence normal (The identified subgroup used to form a central product is central, hence normal).

[L4]

The canonical maps into a central product are injective homomorphisms whose images commute elementwise, generate the product, and meet in the image of the identified subgroup (The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre).

[L5]

For a finite group $G$ and $H \leq G$ , $∣ G ∣ = [G : H] ∣ H ∣$ (Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

[L6]

$∣ A ∣$ is the unique natural number $n$ with $A \approx n$ (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct

1.1

Since $P_{i} / Z (P_{i})$ is elementary abelian, $g^{2} \in Z (P_{i}) = {1, z_{i}}$ for every $g \in P_{i}$ ; so $P_{i}$ splits into the $t (P_{i})$ elements with $g^{2} = 1$ and the $∣ P_{i} ∣ - t (P_{i})$ elements with $g^{2} = z_{i}$ .

F4L1L2

1.2

The quotient map $π$ is surjective with kernel $N = {(z, α (z)^{- 1}) : z \in Z (P_{1})}$ , which has two elements, so every element of $P$ has exactly two preimages in $P_{1} \times P_{2}$ .

F1F3L3L5L6

1.3

Because the two canonical images commute, $π (g, h)^{2} = π (g^{2}, h^{2})$ for all $g \in P_{1}$ and $h \in P_{2}$ .

F2F3L4

2.1

Hence $π (g, h)^{2} = 1$ exactly when $(g^{2}, h^{2}) \in N$ , that is exactly when $g^{2} = 1$ and $h^{2} = 1$ , or $g^{2} = z_{1}$ and $h^{2} = α (z_{1})^{- 1} = z_{2}$ .

F1F4step 1.1step 1.3

3.1

The number of pairs $(g, h)$ with $π (g, h)^{2} = 1$ is therefore $t (P_{1}) t (P_{2}) + (∣ P_{1} ∣ - t (P_{1})) (∣ P_{2} ∣ - t (P_{2}))$ .

L6step 1.1step 2.1

4.1

Each element of $P$ with square the identity is counted exactly twice in that total, so $t (P)$ is half of it, which is the displayed formula.

L6step 1.2step 3.1∎

Remarks

The second summand is what makes the formula more than a product: an element of $P$ can square to the identity because both of its coordinates square to the identity, or because both square to the identified central element and those two squares cancel in the quotient.

Writing $∣ P_{i} ∣ = 2^{1 + 2 n_{i}}$ and $t (P_{i}) = 2^{2 n_{i}} + ε_{i} 2^{n_{i}}$ with $ε_{i} = \pm 1$ , the formula collapses to $t (P) = 2^{2 n} + ε_{1} ε_{2} 2^{n}$ with $n = n_{1} + n_{2}$ : the signs multiply.

Depends on

Used by

Dependency tree · two levels

48 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. A. Craven, The Theory of p-Groups, Theorem 3.14(ii) (standard reference, not scraped)
D. Kaur and A. Kulshrestha, Characters of real special 2-groups, Remark 2.4 (standard reference, not scraped)

A product formula for the number of square roots of the identity in a central product of extraspecial 22-groups