ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-28

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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 finite abelian group Z/12 has length three

Example

The abelian group $Z / 12$ has finite length $3$ . For the subobject $B = ⟨ 2 ⟩ ≅ Z / 6$ , one has $ℓ (B) = 2$ and $ℓ ((Z / 12) / B) = 1$ , so the additivity formula reads $3 = 2 + 1$ .

Facts & Assumptions

Given: The abelian group $A = Z / 12$ and its subgroup $B = ⟨ 2 ⟩$ .

[L1]

Jordan-Holder makes the length independent of the chosen composition series (Jordan-Holder theorem in an abelian category).

[L2]

Finite length and length are the notions of Object of finite length.

[L3]

Length is additive along a subobject (Length is additive along a subobject).

Verification

technique · direct

1.1L1L2algebra

The chain $0 < ⟨ 6 ⟩ < ⟨ 2 ⟩ < Z / 12$ is a composition series of $A$ , so $A$ has finite length and $ℓ (A) = 3$ by [L1] and [L2]. The subgroup $B = ⟨ 2 ⟩ ≅ Z / 6$ has composition series $0 < ⟨ 6 ⟩ < ⟨ 2 ⟩$ , so $ℓ (B) = 2$ , while $A / B ≅ Z / 2$ has length $1$ .

2.1L3step 1.1∎

The numerical identity $ℓ (A) = ℓ (B) + ℓ (A / B)$ becomes $3 = 2 + 1$ in this case, exactly as [L3] asserts.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)