CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A nontrivial finite abelian group is cyclic if and only if it has one invariant factor

Statement

A nontrivial finite abelian group is cyclic if and only if its invariant-factor list has exactly one entry.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

For every finite abelian group $G$ there is a unique list $1 < n_{1} ∣ \dots ∣ n_{r}$ such that $G ≅ C_{n_{1}} \times \dots \times C_{n_{r}}$ . Moreover $∣ G ∣ = n_{1} \dots n_{r}$ . The trivial group corresponds to the empty list and empty product. (Fundamental theorem of finite abelian groups: invariant-factor form).

[L2]

If $G ≅ C_{n_{1}} \times \dots \times C_{n_{r}}$ with $1 < n_{1} ∣ \dots ∣ n_{r}$ , then $∣ G ∣ = n_{1} \dots n_{r}$ and $\exp (G) = n_{r}$ . For the empty list, $∣ G ∣ = \exp (G) = 1$ . (Invariant factors determine the order and exponent of a finite abelian group).

[L3]

If $G = ⟨ g ⟩$ is cyclic, then exactly one of the following applies: - if $g$ has infinite order, $G ≅ (Z, +)$ ; - if $g$ has finite order $n$ , necessarily $n \geq 1$ , then $G ≅ (Z / n, +)$ . (Every cyclic group is isomorphic to $(Z, +)$ or to $(Z / n, +)$ for its finite order $n \geq 1$ ).

Proof

technique · direct

1.1

A one-entry invariant-factor decomposition is an isomorphism with one cyclic group, so $G$ is cyclic.

givenL1L2L3

2.1

Conversely a nontrivial finite cyclic group is isomorphic to $C_{∣ G ∣}$ , giving the one-entry list; uniqueness of invariant factors rules out any different list.

step 1.1∎

Depends on

Used by

Every finite subgroup of the unit group of an integral domain is cyclic Corollary

Dependency tree · two levels

17 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

Keith Conrad, Decomposition of Finite Abelian Groups, §§1-4 (standard reference, not scraped)
Richard Elman, Lectures on Abstract Algebra, Ch. 14 (standard reference, not scraped)