TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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

Statement

Let $P$ be a finite $p$ -group and let $N\mathrel{\trianglelefteq}P$ be nontrivial. Then

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

Facts & Assumptions

Given: A finite $p$ -group $P$ and a nontrivial normal subgroup $N\mathrel{\trianglelefteq}P$ .

[L1]

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

[L2]

A finite $p$ -group action satisfies the fixed-point congruence (If a finite $p$ -group $P$ acts on a finite set $X$ , then $|X|\equiv|X^P|\pmod p$ ).

[L3]

Normality means $gNg^{-1}=N$ 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 $x\mapsto gxg^{-1}$ 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 $k\ge1$ (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

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 86 results over 21 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

K. Conrad, Group Actions, Theorem 5.3 (standard reference, not scraped)

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