PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-26

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Every conjugacy class of an extraspecial $p$ -group outside the centre has exactly $p$ elements

Statement

Every conjugacy class of an extraspecial $p$ -group whose representative is not central has exactly $p$ elements. That is, for an extraspecial $p$ -group $P$ and $x \in P ∖ Z (P)$ ,

$∣ {Cl}_{P} (x) ∣ = p .$

Facts & Assumptions

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

[F1]

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

[F2]

$Z (G) : = {z \in G : z g = g z for every g \in G}$ (The center $Z (G)$ of a group).

[F3]

For $g, h \in G$ the commutator is $[g, h] : = g h g^{- 1} h^{- 1}$ (Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$ ).

[F4]

For $H \leq G$ and $g \in G$ , the right coset is $H g : = {h g : h \in H}$ (Left and right cosets $g H$ and $H g$ of a subgroup).

[L1]

An extraspecial $p$ -group has derived subgroup $P^{'} = Z (P)$ of order $p$ (An extraspecial $p$ -group is nilpotent of class exactly two and its derived subgroup has order $p$ ).

[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 $G$ ).

[L4]

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

[L5]

A finite $p$ -group is a finite group whose order has the form $∣ P ∣ = p^{n}$ for some $n \in N$ (A finite $p$ -group has order $p^{n}$ for a prime $p$ and some $n \in N$ ).

Proof

technique · direct

1.1F3algebra

For every $g \in P$ one has $[g, x] x = (g x g^{- 1} x^{- 1}) x = g x g^{- 1}$ , so each conjugate of $x$ has the form $[g, x] x$ .

1.2L1L5

The derived subgroup of $P$ is $Z (P)$ and has order $p$ , and $P$ is a finite $p$ -group, so $∣ P ∣ = p^{n}$ for some $n$ .

1.3F1L2L3L4

The size of the class of $x$ is the index $[P : C_{P} (x)]$ , and by Lagrange that index divides $∣ P ∣$ .

2.1F4step 1.1step 1.2algebra

Each $[g, x]$ lies in $P^{'} = Z (P)$ , so every conjugate of $x$ lies in the right coset $Z (P) x$ ; the map $z \mapsto z x$ is a bijection from $Z (P)$ onto that coset, so the coset has $p$ elements and the class of $x$ has at most $p$ .

2.2F1F2step 1.3

Since $x \notin Z (P)$ , some $g \in P$ fails to commute with $x$ , so $C_{P} (x) \neq P$ , its index is greater than one, and the class of $x$ has more than one element.

3.1step 1.2step 1.3step 2.1step 2.2∎

The class size divides $p^{n}$ , so it is a power of $p$ ; it lies strictly between $1$ and $p$ inclusive, and the only such power of $p$ is $p$ itself.

Remarks

The hypothesis $x \notin Z (P)$ is used only at step 2.2, and it is used to rule out the class of size one. A central $x$ runs through the same computation and comes out with the class ${x}$ , which is consistent with step 2.1 and shows that the two cases exhaust the group.

Depends on

Used by

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

Dependency tree · two levels

38 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)
D. A. Craven, The Theory of p-Groups, §3.1 (standard reference, not scraped)

Every conjugacy class of an extraspecial pp-group outside the centre has exactly pp elements