CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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An extraspecial $p$ -group has order $p^{1 + 2 n}$ for some $n \geq 1$

Statement

Let $P$ be an extraspecial $p$ -group. Then $∣ P ∣ = p^{1 + 2 n}$ for some integer $n \geq 1$ , and $∣ P / Z (P) ∣ = p^{2 n}$ . In particular no extraspecial group has order $p^{2 m}$ , and none has order $p$ .

Facts & Assumptions

Given: An extraspecial $p$ -group $P$ .

[F1]

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

[L1]

Every extraspecial $p$ -group is an internal central product of $n \geq 1$ nonabelian subgroups of order $p^{3}$ with pairwise intersections $Z (P)$ , and $∣ P ∣ = p^{1 + 2 n}$ (Every extraspecial $p$ -group is an internal central product of nonabelian subgroups of order $p^{3}$ ).

[L2]

For a finite $p$ -group $P$ the following are equivalent: $P$ is extraspecial; $P$ is nonabelian, $∣ Z (P) ∣ = p$ and $P / Z (P)$ is elementary abelian; $P$ is nonabelian and $Z (P) = P^{'} = Φ (P)$ has order $p$ (Three equivalent descriptions of an extraspecial $p$ -group).

[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).

Proof

technique · direct

1.1

The decomposition theorem writes $P$ as an internal central product of $n \geq 1$ nonabelian subgroups of order $p^{3}$ and gives $∣ P ∣ = p^{1 + 2 n}$ .

1.2

The centre of $P$ has order $p$ .

F1L2

2.1

Lagrange applied to $Z (P) \leq P$ gives $∣ P / Z (P) ∣ = ∣ P ∣ / p = p^{2 n}$ .

L3L4step 1.1step 1.2

3.1

Since $1 + 2 n$ is odd, no extraspecial group has order an even power of $p$ ; and $n \geq 1$ excludes the order $p$ , which corresponds to $n = 0$ .

step 1.1step 2.1∎

Remarks

The exponent $n$ is determined by the order and therefore by the group, so it can be used as an invariant even though the decomposition producing it is not unique. That $n \geq 1$ is what the nonabelian clause of the definition buys: an abelian group with a centre of order $p$ would be the cyclic group of order $p$ , of order $p^{1 + 2 \cdot 0}$ .

Depends on

Used by

Dependency tree · two levels

41 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, Theorem 2.40(i) and Proposition 2.41(i) (standard reference, not scraped)
D. A. Craven, The Theory of p-Groups, Theorem 3.9 (standard reference, not scraped)

An extraspecial pp-group has order p1+2np1+2n for some n≥1n≥1