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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The Heisenberg group of order $27$ has exponent $3$ and thirteen subgroups of order $3$

Example

The Heisenberg group of order $27$ has exponent $3$ and thirteen subgroups of order $3$ .

Facts & Assumptions

Given: The objects and hypotheses in the Example.

[F1]

The Heisenberg group of order $p^{3}$ is the set $(Z / p)^{3}$ with $(a, b, c) (a^{'}, b^{'}, c^{'}) = (a + a^{'}, b + b^{'}, c + c^{'} + a b^{'})$ (The Heisenberg group of order $p^{3}$ over $Z / p$ ).

[L1]

The Heisenberg multiplication makes $(Z / p)^{3}$ a nonabelian group of order $p^{3}$ (The Heisenberg multiplication is a group law, nonabelian, on a set of $p^{3}$ elements).

[L2]

The Heisenberg group of order $p^{3}$ is extraspecial, and for odd $p$ it has exponent $p$ (The Heisenberg group of order $p^{3}$ is extraspecial, and for odd $p$ it has exponent $p$ ).

[L3]

For a finite group $G$ , its exponent is $\exp (G) = \min {n \in N : n > 0 and g^{n} = e for every g \in G} .$ The set is nonempty by, and gives its least member; powers use. (The exponent of a finite group).

[L4]

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).

[L5]

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

Verification

technique · direct

1.1F1L1

Instantiate the multiplication at $p = 3$ to obtain a nonabelian group of order twenty-seven.

1.2L2L3

Every nonidentity element cubes to the identity, since three is odd and the general exponent statement applies.

2.1L4L5step 1.2algebra∎

The twenty-six nonidentity elements therefore fall into thirteen subgroups of order three, each containing two of them.

Depends on

Used by

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Dependency tree · two levels

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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 Heisenberg group of order 2727 has exponent 33 and thirteen subgroups of order 33