DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$F_{p}$ -spanning sets, independence, and bases in an elementary abelian $p$ -group

Definition

Let $E$ be an elementary abelian $p$ -group with the canonical scalar action of An elementary abelian $p$ -group has a canonical $F_{p}$ -vector-space structure.

A subset $S \subseteq E$ spans $E$ when every $x \in E$ can be written as a finite product

$x = \prod_{s \in S} s^{a_{s}}, a_{s} \in F_{p},$

with all but finitely many coefficients zero. Equivalently, $⟨ S ⟩ = E$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

The set $S$ is independent when $\prod_{s \in S} s^{a_{s}} = e$ with finite support forces every $a_{s} = 0$ . A basis of an elementary abelian $p$ -group is an independent spanning subset for its canonical $F_{p}$ -linear structure.

The empty subset is independent. It spans exactly the trivial group, so the trivial group has the empty basis.

Depends on

Used by

Dependency tree · two levels

10 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

K. Conrad, Generating Sets, §6 (standard reference, not scraped)
D. A. Craven, The Theory of p-Groups, §2.2 (standard reference, not scraped)

FpFp​-spanning sets, independence, and bases in an elementary abelian pp-group