CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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A noncentral element of an extraspecial $p$ -group has centraliser of index $p$

Statement

Let $P$ be an extraspecial $p$ -group and let $x \in P ∖ Z (P)$ . Then

$[P : C_{P} (x)] = p, equivalently ∣ C_{P} (x) ∣ = \frac{∣ P ∣}{p} .$

Facts & Assumptions

Given: An extraspecial $p$ -group $P$ and an element $x \in P$ with $x \notin Z (P)$ .

[F1]

$C_{G} (x) : = {g \in G : g x = x g} = {g \in G : g x g^{- 1} = x}$ (The conjugacy class ${Cl}_{G} (x)$ and centralizer $C_{G} (x)$ of an element).

[F2]

$[G : H] : = ∣ G / H ∣$ , the number of left cosets of $H$ in $G$ (The coset set $G / H$ and the index $[G : H]$ of a subgroup).

[L1]

Every conjugacy class of an extraspecial $p$ -group whose representative is not central has exactly $p$ elements (Every conjugacy class of an extraspecial $p$ -group outside the centre has exactly $p$ elements).

[L2]

$∣ {Cl}_{G} (x) ∣ = [G : C_{G} (x)]$ for a finite group $G$ ( $G / C_{G} (x) \to {Cl}_{G} (x)$ is a bijection, so $∣ {Cl}_{G} (x) ∣ = [G : C_{G} (x)]$ whenever these cardinalities are finite).

[L3]

For a finite group $G$ and $H \leq G$ , $∣ G ∣ = [G : H] ∣ H ∣$ (Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

Proof

technique · direct

1.1

The class of $x$ has exactly $p$ elements, because $x$ is not central.

1.2

The size of that class is the index of the centraliser of $x$ in $P$ .

F1F2L2

2.1

Hence $[P : C_{P} (x)] = p$ , and Lagrange turns this into $∣ P ∣ = p ∣ C_{P} (x) ∣$ .

L3step 1.1step 1.2∎

Remarks

Every centraliser named here is proper, since $x$ is noncentral, and maximal in the order sense: index $p$ is the smallest index a proper subgroup of a finite $p$ -group can have.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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

M. van Beek, Topics in Finite p-Groups, Proposition 2.41(ii) (standard reference, not scraped)

A noncentral element of an extraspecial pp-group has centraliser of index pp