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

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The commutator pairings of $Dih (C_{4})$ and $Q_{8}$ are the same, while the groups are not isomorphic

Example

The commutator pairings of $Dih (C_{4})$ and $Q_{8}$ are the same, while the groups are not isomorphic.

Facts & Assumptions

Given: The objects and hypotheses in the Example.

[F1]

For an extraspecial $p$ -group $P$ with $Z (P) = ⟨ z ⟩$ , the commutator pairing is the map $b_{z} (\overset{ˉ}{x}, \overset{ˉ}{y}) \in Z / p$ determined by $[x, y] = z^{b_{z} (\overset{ˉ}{x}, \overset{ˉ}{y})}$ (The commutator pairing of an extraspecial $p$ -group relative to a chosen generator of its centre).

[L1]

The commutator pairing is independent of the coset representatives, is $F_{p}$ -bilinear on $P / Z (P)$ , and is alternating (The commutator pairing is well defined on the central quotient, is bilinear over $F_{p}$ , and is alternating).

[L2]

The generalized dihedral group $Dih (C_{4})$ and the quaternion group $Q_{8}$ are extraspecial of order $8$ , with six and two solutions of $x^{2} = 1$ respectively ( $Dih (C_{4})$ and $Q_{8}$ are extraspecial of order $8$ , with six and two solutions of $x^{2} = 1$ respectively).

[L3]

$Dih (A) = A ⋊ C_{2} .$ ( The generalized dihedral group $Dih (A) = A ⋊ C_{2}$ for an abelian group $A$ ).

[L4]

$Q_{8} : = {1, - 1, i, - i, j, - j, k, - k} \subseteq H^{\times} .$ (The quaternion group $Q_{8} = {\pm 1, \pm i, \pm j, \pm k}$ inside the nonzero quaternions).

[L5]

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G \approx n$ for some $n \in N$ . (The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity).

Verification

technique · direct

1.1F1L1L2L3algebra

In $Dih (C_{4})$ the images of $r$ and $s$ form a basis of the central quotient and $[r, s] = r^{2}$ , so the pairing sends that basis pair to $1$ .

2.1F1L4step 1.1algebra

In $Q_{8}$ the images of $i$ and $j$ form a basis and $[i, j] = - 1$ , so the pairing again sends that basis pair to $1$ ; the two pairings agree in these coordinates.

3.1L2L5step 2.1∎

The groups are nevertheless not isomorphic, because six elements satisfy $x^{2} = 1$ in the first and two in the second.

Depends on

Used by

FALSE: two extraspecial p-groups whose commutator pairings agree are isomorphic False statement

Dependency tree · two levels

38 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)

The commutator pairings of Dih⁡(C4)Dih(C4​) and Q8Q8​ are the same, while the groups are not isomorphic