DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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$p$ -elementary and $p$ -hyperelementary finite groups

Definition

Let $p$ be prime. A finite group is $p$ -elementary if it is isomorphic to $C \times P$ , where $C$ is cyclic of order prime to $p$ and $P$ is a finite $p$ -group. It is $p$ -hyperelementary (also called $p$ -quasi-elementary) if it is isomorphic to $C ⋊ P$ with the same conditions. The trivial group is allowed for either factor. The family of elementary subgroups means the union of the $p$ -elementary families over all primes.

Depends on

Used by

Brauer induction for S₃ Example
Small elementary and hyperelementary groups Example
Trivial factors in an elementary group Example
Characters of elementary groups are induced from linear characters Lemma
Elementary detection at a fixed element Lemma
Elementary groups are supersolvable Lemma
Hyperelementary permutation subring reduction Lemma
Isaacs' linear-character step Lemma
p-primary congruence for integer-valued cyclotomic character combinations Lemma
Subgroups of elementary and hyperelementary groups Lemma
Brauer induction Theorem

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

Wen-Wei Li, Yanqi Lake Lectures on Algebra I, Definition 14.1.1 (standard reference, not scraped)

pp-elementary and pp-hyperelementary finite groups