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A central product of extraspecial $p$ -groups identified along their centres is extraspecial

Statement

Let $E_{1}$ and $E_{2}$ be extraspecial $p$ -groups and let $α : Z (E_{1}) \to Z (E_{2})$ be an isomorphism. Then $P = E_{1} \circ_{α} E_{2}$ is extraspecial, of order $∣ E_{1} ∣ ∣ E_{2} ∣ / p$ , and its centre and derived subgroup are the common image of $Z (E_{1})$ and $Z (E_{2})$ .

Facts & Assumptions

Given: Extraspecial $p$ -groups $E_{1}, E_{2}$ , an isomorphism $α : Z (E_{1}) \to Z (E_{2})$ , and $P = E_{1} \circ_{α} E_{2}$ with canonical images $\overset{ˉ}{g}$ for $g \in E_{1}$ and $\overset{ˉ}{h}$ for $h \in E_{2}$ .

[F1]

A finite $p$ -group $P$ is special when $Z (P) = P^{'} = Φ (P)$ is elementary abelian, and extraspecial when in addition $P$ is nonabelian and this common subgroup has order $p$ (Special and extraspecial $p$ -groups).

[F2]

A finite $p$ -group is a finite group whose order has the form $∣ P ∣ = p^{n}$ for some $n \in N$ (A finite $p$ -group has order $p^{n}$ for a prime $p$ and some $n \in N$ ).

[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]

For a central product of finite groups, $∣ G \circ_{α} H ∣ = ∣ G ∣ ∣ H ∣ / ∣ Z_{1} ∣$ ; without a finiteness hypothesis, its centre is the image of $Z (G) \times Z (H)$ and its derived subgroup is the image of $[G, G] \times [H, H]$ (Order, centre and derived subgroup of a central product).

[L3]

The canonical maps $G \to G \circ_{α} H$ and $H \to G \circ_{α} H$ are injective homomorphisms whose images commute elementwise, generate $G \circ_{α} H$ , and meet in the image of $Z_{1}$ (The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre).

[L4]

For $N ⊴ G$ , the quotient $G / N$ is abelian if and only if $[G, G] \subseteq N$ ( $G / N$ is abelian if and only if $[G, G] \subseteq N$ ).

[L5]

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

Proof

technique · direct

1.1

Each $E_{i}$ is nonabelian, has $Z (E_{i}) = [E_{i}, E_{i}]$ of order $p$ , and has elementary abelian central quotient $E_{i} / Z (E_{i})$ .

1.2

The canonical maps embed $E_{1}$ and $E_{2}$ in $P$ ; their images commute elementwise and generate $P$ .

2.1

The central-product formulas give $∣ P ∣ = ∣ E_{1} ∣ ∣ E_{2} ∣ / p$ , which is a power of $p$ ; the centre of $P$ is the image of $Z (E_{1}) \times Z (E_{2})$ and the derived subgroup of $P$ is the image of $[E_{1}, E_{1}] \times [E_{2}, E_{2}]$ . Those two subgroups of $E_{1} \times E_{2}$ coincide by step 1.1, so $Z (P) = [P, P]$ ; the subgroup $Z (E_{1}) \times Z (E_{2})$ has order $p^{2}$ and contains the identified subgroup of order $p$ , so its image has order $p$ .

F2L2step 1.1algebra

2.2

The group $P$ is nonabelian, because the canonical map embeds the nonabelian group $E_{1}$ into it.

step 1.1step 1.2

3.1

Since $[P, P] = Z (P)$ , the quotient $P / Z (P)$ is abelian; and for $g \in E_{1}$ , $h \in E_{2}$ the commuting images give $(\overset{ˉ}{g} \overset{ˉ}{h})^{p} = {\overset{ˉ}{g}}^{p} {\overset{ˉ}{h}}^{p} = \overline{g^{p}} \overline{h^{p}}$ , where $g^{p} \in Z (E_{1})$ and $h^{p} \in Z (E_{2})$ because the central quotients are elementary abelian, so $(\overset{ˉ}{g} \overset{ˉ}{h})^{p}$ lies in $Z (P)$ . Every element of $P$ has this form, so every element of the finite abelian $p$ -group $P / Z (P)$ has order dividing $p$ and $P / Z (P)$ is elementary abelian.

L4L5step 1.1step 1.2step 2.1

4.1

So $P$ is a nonabelian finite $p$ -group with $∣ Z (P) ∣ = p$ and elementary abelian central quotient, which is the second description in the characterisation; hence $P$ is extraspecial.

F1L1step 2.1step 2.2step 3.1∎

Remarks

The identification must be along the full centres: if a proper subgroup of $Z (E_{1})$ were identified it would be trivial, the product would be the direct product, and its centre would have order $p^{2}$ . That is the case recorded on the companion page as a special group which is not extraspecial.

Depends on

Used by

Dependency tree · two levels

37 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

M. van Beek, Topics in Finite p-Groups, Propositions 2.36 and 2.39(i) (standard reference, not scraped)
D. A. Craven, The Theory of p-Groups, §3.3 (standard reference, not scraped)

A central product of extraspecial pp-groups identified along their centres is extraspecial