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An automorphism of an extraspecial $p$ -group acting trivially on its Frattini quotient is inner

Statement

Let $P$ be an extraspecial $p$ -group of order $p^{1 + 2 n}$ and let $ρ_{P} : Aut (P) \to {Aut}_{F_{p}} (P / Φ (P))$ be the induced-action homomorphism. Then $\ker ρ_{P} = Inn (P)$ : an automorphism of $P$ acting trivially on the Frattini quotient is inner.

Facts & Assumptions

Given: An extraspecial $p$ -group $P$ of order $p^{1 + 2 n}$ .

[F1]

$Inn (G) : = {c_{g} : g \in G}$ with $c_{g} (x) = g x g^{- 1}$ (Inner automorphisms and $Inn (G)$ ).

[F2]

For a finite $p$ -group $P$ , the generator rank $d (P)$ is the common size of a basis of $P / Φ (P)$ (The generator rank $d (P)$ of a finite $p$ -group).

[F3]

$Φ (G)$ is the intersection of the maximal proper subgroups of $G$ (The Frattini subgroup $Φ (G)$ as the intersection of the maximal subgroups of a finite group).

[F4]

An isomorphism is a bijective group homomorphism, and $Aut (G)$ is the set of automorphisms of $G$ (Group isomorphisms, automorphisms and the set $Aut (G)$ ).

[L1]

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).

[L2]

An extraspecial $p$ -group has $∣ P ∣ = p^{1 + 2 n}$ with $n \geq 1$ and $∣ P / Z (P) ∣ = p^{2 n}$ (An extraspecial $p$ -group has order $p^{1 + 2 n}$ for some $n \geq 1$ ).

[L3]

An extraspecial $p$ -group of order $p^{1 + 2 n}$ has $Φ (P) = Z (P)$ of order $p$ and generator rank $d (P) = 2 n$ , and every minimal generating set has $2 n$ elements (An extraspecial $p$ -group of order $p^{1 + 2 n}$ has generator rank $2 n$ ).

[L4]

Every automorphism of a finite $p$ -group induces an $F_{p}$ -linear automorphism of $P / Φ (P)$ , and these form a homomorphism $ρ_{P}$ (Automorphisms act linearly on the Frattini quotient).

[L5]

A subset $X$ of a finite $p$ -group $P$ is a minimal generating set if and only if the quotient map restricts to a bijection from $X$ onto a basis of $P / Φ (P)$ (Burnside Basis Theorem).

[L6]

$G / Z (G) ≅ Inn (G)$ ( $G / Z (G) ≅ Inn (G)$ ).

[L7]

If $A$ and $B$ are finite then $A \times B$ is finite and $∣ A \times B ∣ = ∣ A ∣ \cdot ∣ B ∣$ (The product rule: $∣ A \times B ∣ = ∣ A ∣ ∣ B ∣$ , and $∣ \prod_{i < m} A_{i} ∣ = \prod_{i < m} ∣ A_{i} ∣$ ).

[L8]

$∣ A ∣$ is the unique natural number $n$ with $A \approx n$ (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct

1.1F2F3L1L2L3

The Frattini subgroup is $Φ (P) = Z (P)$ of order $p$ , the Frattini quotient has order $p^{2 n}$ , and every minimal generating set of $P$ has exactly $2 n$ elements.

2.1F4L5step 1.1

Fix a minimal generating set $X = {g_{1}, \dots, g_{2 n}}$ , which exists because the Burnside basis theorem matches minimal generating sets with bases of the Frattini quotient. An automorphism of $P$ is determined by its values on $X$ , since $X$ generates $P$ .

2.2F1L1L4step 1.1

Every inner automorphism lies in $\ker ρ_{P}$ : for $g, x \in P$ one has $c_{g} (x) x^{- 1} = g x g^{- 1} x^{- 1} \in [P, P] = Φ (P)$ , so $c_{g}$ fixes every coset of $Φ (P)$ .

2.3F1L2L6step 1.1

The inner automorphism group has order $∣ P / Z (P) ∣ = p^{2 n}$ .

3.1F3L4L7L8step 1.1step 2.1

If $θ$ lies in $\ker ρ_{P}$ then $θ (g_{i}) Φ (P) = g_{i} Φ (P)$ for each $i$ , so $θ (g_{i}) = g_{i} u_{i}$ with $u_{i} \in Φ (P) = Z (P)$ , a set of $p$ elements. Hence $θ$ is determined by the tuple $(u_{1}, \dots, u_{2 n})$ , and there are at most $p^{2 n}$ such tuples, so $∣ \ker ρ_{P} ∣ \leq p^{2 n}$ .

4.1L8step 3.1step 2.2step 2.3∎

So $Inn (P)$ is a subset of $\ker ρ_{P}$ of size $p^{2 n}$ , while $\ker ρ_{P}$ has at most $p^{2 n}$ elements; hence the two coincide.

Remarks

The equality is forced by two counts that happen to agree, and each uses the extraspecial hypothesis: the upper bound uses $Φ (P) = Z (P)$ of order $p$ together with the generator rank $2 n$ , and the lower bound uses $∣ P / Z (P) ∣ = p^{2 n}$ . For a general finite $p$ -group the kernel can be larger than the inner automorphism group; equality is not asserted or excluded without additional hypotheses.

Depends on

Used by

An automorphism fixing the centre pointwise induces a pairing-preserving automorphism of the central quotient, with kernel the inner automorphisms Proposition

Dependency tree · two levels

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Sources

D. A. Craven, The Theory of p-Groups, Theorem 3.15(i) (standard reference, not scraped)

An automorphism of an extraspecial pp-group acting trivially on its Frattini quotient is inner