DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 $G (p) = {g \in G : g^{p^{k}} = e for some k \in N} .$ Thus the identity is included by $k = 0$ . In additive notation, $G (p) = {g : p^{k} g = 0 for some k \in N}$ . 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 $ord (g)$ of an element, with $ord (g) = \infty$ 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 · two levels

19 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)