LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

p-regularity is preserved by conjugacy and by powers coprime to the element order

Statement

Let $x \in G$ and let $p$ be a prime.

If $y = g x g^{- 1}$ for some $g \in G$ , then $x$ is $p$ -regular if and only if $y$ is $p$ -regular.
If $m$ is coprime to $∣ x ∣$ , then $x$ is $p$ -regular if and only if $x^{m}$ is $p$ -regular.

Facts & Assumptions

Given: A finite group $G$ , a prime $p$ , and an element $x \in G$ .

[F1]

An element is $p$ -regular exactly when the prime $p$ does not divide its order (p-regular and p-singular elements).

Proof

technique · direct

1.1givenalgebra

Conjugation preserves order: if $y = g x g^{- 1}$ , then $y^{n} = 1$ exactly when $g x^{n} g^{- 1} = 1$ , that is, exactly when $x^{n} = 1$ . So $∣ y ∣ = ∣ x ∣$ .

1.2givenalgebra

Let $n = ∣ x ∣$ and suppose $\gcd (m, n) = 1$ . Then the cyclic subgroup $⟨ x^{m} ⟩$ equals $⟨ x ⟩$ , because some integer $r$ satisfies $m r \equiv 1 (m o d n)$ , hence $x = (x^{m})^{r}$ . Therefore $∣ x^{m} ∣ = ∣ x ∣$ .

2.1F1step 1.1step 1.2∎

Step 1.1 and [F1] prove claim 1, while step 1.2 and [F1] prove claim 2.

Depends on

p-regular and p-singular elements

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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

J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)