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

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The subobject lattice of a cyclic group of order twelve

Example

For the cyclic group $C_{12}$ , subgroups correspond to divisors of $12$ . The meet of two subgroups is their intersection, corresponding to the gcd of the two orders, and the join is their sum, corresponding to the lcm. In particular this subobject lattice is distributive, unlike the $M_{3}$ witness on the A page.

Facts & Assumptions

Given: The cyclic group $C_{12}$ .

[L1]

Subobjects of an abelian-category object form a lattice (The subobjects of an object in an abelian category form a lattice).

[L2]

Such lattices are modular (The subobject lattice of an abelian category is modular).

Verification

technique · direct

1.1L1algebra

For each divisor $d$ of $12$ , the cyclic group $C_{12}$ has a unique subgroup of order $d$ , namely $⟨ 12 / d ⟩$ . So the subgroup lattice is the divisor lattice of $12$ with elements of orders $1, 2, 3, 4, 6, 12$ . The meet is intersection, hence gcd of orders, and the join is subgroup sum, hence lcm of orders.

2.1L2step 1.1∎

The divisor lattice of a single integer is distributive, so this example is more rigid than the modular-only situation of [L2]. It therefore illustrates that modularity does not force every concrete subobject lattice to look like the $M_{3}$ example.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Daniel Murfet, Abelian Categories, Section 4.2 (standard reference, not scraped)