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

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Two composition series of Z/12 refine to the same simple factors

Example

In the $Z$ -module $Z / 12$ , the chains

$0 < ⟨ 6 ⟩ < ⟨ 2 ⟩ < Z / 12, 0 < ⟨ 4 ⟩ < ⟨ 2 ⟩ < Z / 12$

are two composition series. Their successive factors are $Z / 2, Z / 3, Z / 2$ and $Z / 3, Z / 2, Z / 2$ , so the same simple factors occur up to permutation.

Facts & Assumptions

Given: The module $Z / 12$ .

[L1]

The butterfly lemma is the cell-by-cell quotient comparison behind refinements (Zassenhaus butterfly lemma in an abelian category).

[L2]

Any two finite subobject chains admit equivalent refinements (Schreier refinement theorem in an abelian category).

[L3]

Jordan-Holder identifies the factor multiset up to permutation (Jordan-Holder theorem in an abelian category).

Verification

technique · direct

1.1L1algebra

The subgroup $⟨ 6 ⟩$ has order $2$ and $⟨ 2 ⟩ / ⟨ 6 ⟩$ has order $3$ , while $(Z / 12) / ⟨ 2 ⟩$ has order $2$ . Likewise $⟨ 4 ⟩$ has order $3$ , $⟨ 2 ⟩ / ⟨ 4 ⟩$ has order $2$ , and the top quotient is again order $2$ . So both displayed chains are composition series.

2.1L2L3step 1.1∎

The two series already display the same three simple factors up to permutation, which is the conclusion predicted abstractly by [L2] and [L3]. This is the concrete module calculation that the categorical refinement theorems package.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)