TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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.

Every nontrivial normal subgroup of a finite $p$ -group meets the center nontrivially

Statement

Let $P$ be a finite $p$ -group and let $N ⊴ P$ be nontrivial. Then

$N \cap Z (P) \neq {e} .$

Facts & Assumptions

Given: A finite $p$ -group $P$ and a nontrivial normal subgroup $N ⊴ P$ .

[L1]

A finite $p$ -group has prime-power order (A finite $p$ -group has order $p^{n}$ for a prime $p$ and some $n \in N$ ).

[L2]

A finite $p$ -group action satisfies the fixed-point congruence (If a finite $p$ -group $P$ acts on a finite set $X$ , then $∣ X ∣ \equiv ∣ X^{P} ∣ (m o d p)$ ).

[L3]

Normality means $g N g^{- 1} = N$ for every $g \in P$ (Normal subgroup: invariance under conjugation, Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).

[L4]

Conjugation by $g$ is an automorphism (Conjugation $x \mapsto g x g^{- 1}$ is an automorphism).

[L5]

The center consists of the elements fixed by every conjugation (The center $Z (G)$ of a group).

[L6]

A nontrivial subgroup of a finite $p$ -group has order $p^{k}$ for some $k \geq 1$ (Every subgroup of a finite $p$ -group has order a power of $p$ ).

Proof

technique · direct

1.1

By [L3] and [L4], conjugation restricts to an action of $P$ on the finite set $N$ .

L3L4

2.1

A point of $N$ is fixed by every element of $P$ exactly when it lies in $N \cap Z (P)$ by [L5].

step 1.1L5

3.1

By [L6], $p$ divides $∣ N ∣$ . Applying [L2] to the action in step 1.1 therefore shows that $p$ divides $∣ N \cap Z (P) ∣$ .

step 1.1step 2.1L1L2L6

4.1

The intersection contains $e$ , and its cardinality is a positive multiple of the prime $p > 1$ ; hence it contains a nonidentity element.

step 3.1L5algebra∎

Depends on

Used by

Normal p subgroup has proper commutator in a p group Lemma
Burnside's pᵃqᵇ theorem Theorem

Dependency tree · two levels

23 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, Group Actions, Theorem 5.3 (standard reference, not scraped)

Every nontrivial normal subgroup of a finite pp-group meets the center nontrivially