ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-26

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A choice of four generators exhibiting an extraspecial group of order $32$ as an internal central product

Example

A choice of four generators exhibiting an extraspecial group of order $32$ as an internal central product.

Facts & Assumptions

Given: The central product $Dih (C_{4}) \circ Dih (C_{4})$ and its two canonical factor maps.

[L1]

For $n \geq 1$ , $Dih (C_{n}) = C_{n} ⋊ C_{2}$ has order $2 n$ , and with $C_{n} = ⟨ r ⟩$ and $C_{2} = ⟨ s ⟩$ every element has a unique form $r^{i}$ or $r^{i} s$ with $0 \leq i < n$ ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations).

[L2]

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

[L3]

$⟨ S ⟩ : = ⋂ {H : H \leq G and S \subseteq H} .$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[L4]

$Dih (C_{4})$ is extraspecial of order eight, and a central product of two extraspecial $2$ -groups along their centres is extraspecial of order $∣ E_{1} ∣ ∣ E_{2} ∣ / 2$ ( $Dih (C_{4})$ and $Q_{8}$ are extraspecial of order $8$ , with six and two solutions of $x^{2} = 1$ respectively, A central product of extraspecial $p$ -groups identified along their centres is extraspecial).

Verification

technique · direct

1.1L1L2L3L4

Let $ι_{1}, ι_{2} : Dih (C_{4}) \to Dih (C_{4}) \circ Dih (C_{4})$ be the canonical maps, and put $a : = ι_{1} (r)$ , $b : = ι_{1} (s)$ , $c : = ι_{2} (r)$ and $d : = ι_{2} (s)$ . By the normal form of [L1], the images of the two factors are $⟨ a, b ⟩ = ι_{1} (Dih (C_{4}))$ and $⟨ c, d ⟩ = ι_{2} (Dih (C_{4}))$ , and injectivity shows that each has order eight. The central product is extraspecial and has order $8 \cdot 8 / 2 = 32$ .

2.1L2step 1.1∎

The two subgroups commute elementwise, generate the whole central product, and meet in the common central line. Therefore the four generators $a, b, c, d$ exhibit $Dih (C_{4}) \circ Dih (C_{4})$ as an internal central product of two subgroups of order eight.

Depends on

Used by

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Dependency tree · two levels

33 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 (Hilary Term 2008), 48 pp. (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, 62 pp. (standard reference, not scraped)
D. Kaur and A. Kulshrestha, Characters of real special 2-groups (arXiv:1510.06583v1) (standard reference, not scraped)

A choice of four generators exhibiting an extraspecial group of order 3232 as an internal central product