DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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The p-primary component of an abelian group

Definition

Let $G$ be an abelian group and $p$ a prime. Its $p$ -primary component is $G(p)=\{g\in G:g^{p^k}=e\text{ for some }k\in\mathbb N\}.$ Thus the identity is included by $k=0$ . In additive notation, $G(p)=\{g:p^kg=0\text{ for some }k\in\mathbb N\}$ . Powers and element orders use Powers $g^{n}$ : natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$ and The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity. No finiteness or maximality is part of the definition.

Depends on

Used by

A p-primary component has the full p-power order and is the unique subgroup of that order Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 51 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)