Homology, Homotopy and Applications, vol.??(??), ????, pp.1–19

GROUP EXTENSIONS AND AUTOMORPHISM GROUP RINGS

JOHN MARTINO and STEWART PRIDDY

Abstract We use extensions to study the semi-simple quotient of the group ring F_{p}Aut(P) of a finite p-group P. For an extension E : N → P → Q, our results involve relations between Aut(N), Aut(P), Aut(Q) and the extension class [E] ∈ H^{2}(Q, ZN). One novel feature is the use of the intersection orbit group Ω([E]), defined as the intersection of the orbits Aut(N) · [E] and Aut(Q) · [E] in H^{2}(Q, ZN). This group is useful in computing |Aut(P)|. In case N, Q are elementary Abelian 2-groups our results involve the theory of quadratic forms and the Arf invariant.

1. Introduction

Since the simple modules of a ring and its semi-simple quotient are the same, for many purposes it suffices to consider the latter ring. In this note we study the problem of calculating the semi-simple quotient of the group ring F_{p}Aut(P) for the automorphism group of a finite p-group. The usual method is to consider the maximal elementary Abelian quotient P → P/Φ(P), where Φ(P) is the Frattini subgroup. The induced map Aut(P) → Aut(P/Φ(P)) has a p-group as kernel by the Hall Basis Theorem. Hence the map of algebras

F_{p}Aut(P) → F_{p}Aut(P/Φ(P))

has a nilpotent kernel and thus suffices to compute the semi-simple quotient. However this map is not necessarily onto and one is left with the still considerable problem of determining the image. For Φ(P) = Z/p, when p is an odd prime, this has been done by Dietz [7] giving a complete determination of Aut(P).

We adopt an inductive approach via extensions; that is, we assume P is given as an extension

E : N → P → Q

with Aut(N), Aut(Q) under control. Then there is an exact sequence relating the automorphism groups of N and Q with that of P, depending on the cohomology class of the extension [E] ∈ H^{2}(Q, Z(N)) where Z(N) denotes the center of N.

Our motivation comes from stable homotopy theory. Let G be a finite group and p be a prime. The classifying space of G completed at p, BG_{p}^{∧}, decomposes stably

. 2000 Mathematics Subject Classification: Primary 20J06; Secondary 55P42 Key words and phrases: automorphism group, extension class, semi-simple quotient, stable splittings. \begin{matrix}c \\ ◯ \end{matrix} ????, John Martino and Stewart Priddy. Permission to copy for private use granted.

into a wedge of indecomposable summands

BG_{p}^{∧} ≃ X_{1} ∨ X_{2} ∨ ⋯ ∨ X_{n}.

Each summand X_{i} is the mapping telescope of a primitive idempotent e in the ring of stable self-maps of BG_{p}^{∧}, e ∈ {BG_{p}^{∧}, BG_{p}^{∧}}. Thus there is a one-to-one correspondence between the indecomposable summands and the simple modules of the ring of stable self-maps. This correspondence is explored in both [2] and [9] (see also [10]). It turns out that modular representation theory plays a crucial role: if P is a Sylow p-subgroup of G then each indecomposable summand of BG_{p}^{∧} originates in BQ for some subgroup Q ⩽ P and corresponds to a simple F_{p}Aut(Q) module.

Of course, the automorphism group of a group is of intrinsic interest in its own right, and our methods shed some light on its structure.

An outline of the paper follows: Section 2 covers the preliminaries on extensions, E : N → G → Q, including the fundamental exact sequence, Theorem 2.1, relating Aut(N), Aut(G), Aut(Q), and the extension class [E]. In Section 3 we define and identify a group structure, Ω([E]), on the intersection of the two orbits Aut(N) · [E] and Aut(Q) · [E] where Aut(N) and Aut(Q) act on H^{2}(Q, Z(N)) in the usual way. This group, which we call the intersection orbit group, is useful in computing |Aut(G)|. The case of trivial action (or twisting) of Q on Z(N) is considered in Section 4. Extensions with N, Q, elementary Abelian p-groups are studied in Section 5. In case p = 2 this involves the theory of quadratic forms over F_{2} and the Arf invariant. We recall Browder’s classification theorem [3] and give several results describing the order of a quotient of Aut(G) by a normal p-subgroup in Section 6. For more complicated p-groups G we describe an inductive procedure for extending these results using the mod-p lower central series. Section 7 is devoted to several applications of the theory.

In what follows all groups are assumed finite, except as noted in Section 3.

2. Preliminaries

We begin by recalling the results of C. Wells [14] as extended by J. Buckley [4]. Because their notation is now non-standard, e.g., functions written on the right, we re-couch these results in more standard notation. Let

E : N →^{i} G →^{π} Q

be an extension of the group N by the group Q and let Aut_{N}(G) be the group of automorphisms of G mapping N to itself. The obvious homomorphism ρ = (ρ_{Q}, ρ_{N}) : Aut_{N}(G) → Aut(Q) × Aut(N) provides a means of studying Aut_{N}(G).

As usual two extensions E_{1}, E_{2} are equivalent E_{1} ∼ E_{2} if there is an isomorphism α : G_{1} → G_{2} restricting to the identity on N and inducing the identity on Q. The set of such equivalent extensions is denoted E(Q, N). The twisting (or coupling) χ : Q → Out(N) of E is the homomorphism defined as usual by χ(q)(n) = i^{−1}(g^{−1}i(n)g) where g ∈ π^{−1}(q), n ∈ N. Equivalent extensions have the same twisting.

The center ZN of N has the structure of a Q module via a homomorphism \overline{χ} : Q → Aut(ZN) defined by the composite

\overline{χ} : Q →^{χ} Out(N) →^{res} Aut(ZN)

where res : Out(N) → Aut(ZN) is induced by Aut(N) → Aut(ZN). It is wellknown that we may identify

E(Q, N) = ∐_{χ} H2 ; \overline{χ}(Q, ZN)

where χ ranges over all twistings {Q → Out(N)}.

Now consider (σ, τ) ∈ Aut(Q) × Aut(N) and form the extension

σEτ^{−1} : N →^{iτ^{−1}} G →^{σπ} Q

Then Aut(Q) × Aut(N) acts on E(Q, N) from the left by

(σ, τ)[E] = [σEτ^{−1}].  (1)

One checks (σ, τ)(σ′, τ′)[E] = (σσ′, ττ′)[E] and (1, 1)[E] = [E]. The twisting of (σ, τ)E is given by γ\overline{τ} χσ^{−1} where γ\overline{τ} denotes conjugation by \overline{τ}, the image of τ in Out(N). For a given χ define the subgroup C_{χ} ⊂ Aut(Q) × Aut(N) by

C_{χ} = \{(σ, τ) ∈ Aut(Q) × Aut(N) | γ\overline{τ} χσ^{−1} = χ\}

that is, the following diagram commutes

[figure: Q; χ; Out(N); Q; σ; χ; Out(N); γ\overline{τ}]

The subgroup C_{χ} consists of all ordered pairs (σ, τ) ∈ Aut(Q) × Aut(N) that preserve the twisting.

If χ is trivial then clearly C_{χ} = Aut(Q) × Aut(N). We note that ker(χ) plays no role in the commutativity of the diagram and σ|_{ker(χ)} : ker(χ) → ker(χ). Thus if the sequence

ker(χ) → Q → im(χ)

splits, e.g., if Q is elementary Abelian, then σ|_{ker(χ)} can be an arbitrary linear isomorphism.

Then C_{χ} acts on {[E] | E has twisting χ}. It is trivial to check that Im(ρ) ⊂ C_{χ} so we consider ρ as a homomorphism ρ : Aut_{N}(G) → C_{χ}.

Let Z1 ; \overline{χ}(Q, ZN) denote the group of derivations, i.e., functions f : Q → ZN satisfying f(qq′) = f(q) + qf(q′) for q, q′ ∈ Q. Then there is a homomorphism μ : Z1 ; \overline{χ}(Q, ZN) → Aut_{N}(G) defined by μ(f)(g) = f(π(g)) · g.

Finally we define a function ϵ : C_{χ} → H2 ; \overline{χ}(Q, ZN) by restricting the action of Aut(Q) × Aut(N) on the extension class [E], that is, (σ, τ) ↦ (σ, τ)[E]. In general ϵ is not a homomorphism.

The following is the principal result of [14] as extended by [4]:

Theorem 2.1. For a given extension E : N → G → Q with twisting χ : Q → Out(N) there is an exact sequence

1 → Z1 ; \overline{χ}(Q, ZN) →^{μ} Aut_{N}(G) →^{ρ} C_{χ} →^{ϵ} H2 ; \overline{χ}(Q, ZN)

with Im(ρ) = (C_{χ})_{[E]}, the isotropy subgroup of C_{χ} fixing [E]. The map ϵ is not onto and is only a set map.

An alternate exact sequence results by replacing Z1 ; \overline{χ}(Q, ZN) with the cohomology group H1 ; \overline{χ}(Q, ZN) and Aut_{N}(G) by Aut_{N}(G)/Inn_{ZN}(G) where Inn_{ZN}(G) is group of inner automorphisms of G induced by elements of ZN, (see [12], Proposition IV 2.1). We shall be interested in Im(ρ) so we will not need this refinement.

In case ZN is a p-group Theorem 2.1 shows Z1 ; \overline{χ}(Q, ZN) is a normal p-subgroup of Aut_{N}(G) which in turn implies F_{p}((C_{χ})_{[E]}) suffices to compute simple modules and idempotents:

Corollary 2.2. If E : N → G → Q is an extension with ZN a p-group, then

F_{p}Aut_{N}(G) ⟶^{F_{p}(ρ)} F_{p}((C_{χ})_{[E]}) is surjective with nilpotent kernel.

Proof: Since ker(ρ) = Z1 ; \overline{χ}(Q, ZN) is a p-group it is well known that ker(F_{p}(ρ)) is nilpotent.

We shall be interested in determining when Aut_{N}(G) is a p-group; clearly we have

Corollary 2.3. If Aut(Q) and Aut(N) are p-groups then Aut_{N}(G) is a p-group.

Corollary 2.4. Suppose G is a p-group with p odd, and N is generated by the elements of G of order p. Then Aut(G) is a p-group if Aut(N) is a p-group.

Proof: By hypothesis N = Ω_{1}(G) is a characteristic subgroup, hence Aut_{N}(G) = Aut(G). If α ∈ Aut(G) has p′ order then ρ_{N}(α) = id_{N}. Since p is odd, the automorphisms of G which have p′ order and fix N are trivial by Theorem 5.3.10, [8]. Thus Aut(G) is a p-group.

3. The Intersection Orbit Group

Let X be a left A × B set where A, B are groups, not necessarily finite. Equivalently A acts on the left, B acts on the right such that a(xb) = (ax)b, and (a, b)x = axb^{−1}, where a ∈ A, x ∈ X, b ∈ B. As usual let Ax denote the orbit of x under the action of A (respectively xB denote the orbit of x under the action of B). We define the intersection orbit group at x

Ω(x) := (Ax) ∩ (xB)

If ax = xb, a′x = xb′ are elements of Ω(x), their product is defined by

(ax)(a′x) := (aa′)x = x(bb′)

It is straightforward to check that this pairing is well-defined giving Ω(x) the structure of a group. Although left and right actions commute, Ω(x) is not necessarily Abelian; however, if either A or B is a p-group, then Ω(x) is a p-group. Also if A or B is trivial then obviously Ω(x) = {x}, the trivial group.

Again, as usual, let A_{x} and B_{x} denote the respective isotropy subgroups.

Proposition 3.1. There is an isomorphism of groups

ϕ : (A × B)_{x}/(A_{x} × B_{x}) ⟶^{≅} Ω(x)

given by ϕ(a, b) = ax = xb^{−1}.

Proof: (a, b)x = x if and only if ax = xb^{−1}. Similarly (a, b) ∈ A_{x} × B_{x} if and only if ax = xb^{−1} = x. Thus ϕ is well-defined and bijective. It is a homomorphism by the definition of the product in Ω(x).

Corollary 3.2. If A and B are finite groups, then

|Ω(x)| divides

|N_{A}(A_{x})/A_{x}|, |N_{B}(B_{x})/B_{x}|,

and

gcd(|Ax|, |xB|).

Proof: Note that N_{A}(A_{x})/A_{x} is the largest subset of Ax ≅ A/A_{x} which is a group. Thus Ω(x) ⩽ N_{A}(A_{x})/A_{x} is a subgroup and so |Ω(x)| divides |N_{A}(A_{x})/A_{x}| and hence A/A_{x}. Similarly for B. Thus |Ω(x)| divides gcd(|Ax|, |xB|).

Returning to automorphism groups, the following result often simplifies computing |Im(ρ)|.

Corollary 3.3. For a given extension of finite groups E : N → G → Q, i) |Im(ρ)| = |Aut(Q)_{[E]}| · |Aut(N)_{[E]}| · |Ω([E])| ii) Im(ρ) is a p-group if and only if Aut(Q)_{[E]}, Aut(N)_{[E]}, and |Ω([E])| are

p-groups.

Proof: ii) follows from i) which follows immediately from Proposition 3.1 with A = Aut(Q), B = Aut(N).

There exists extensions that do not depend on the intersection orbit group.

Proposition 3.4. Suppose N is an Abelian p-group, Aut(Q) is a p-group, and H^{2}(Q, N) ≠ 0 with a trivial twisting. Then there exists a non-split extension [E] such that

|Aut_{N}(G)|_{p} = |Hom(Q, N)||Aut(Q)||Aut(N)|_{p}

Proof: By Corollary 3.2, it suffices find [E] with |Ω([E])| = 1. Let Aut(Q)×S be a Sylow p-subgroup of Aut(Q)×Aut(N). The Aut(Q)×Aut(N)-module H^{2}(Q, N) has cardinality a power of p. Thus there is an Aut(Q)×S fixed point, [E] ≠ 0 by the fixed point result [13], p. 64, which generalizes to this case). Thus Aut(Q)_{[E]} = {[E]} and |S| = |Aut(N)|_{p}. The kernel of the exact sequence in Theorem 2.1 is Z^{1}(Q, N) = Hom(Q, N), which explains the presence of that term in the proposition.

4. Trivial twisting

Proposition 4.1. If E : N → G → Q has trivial twisting the exact sequence of Theorem 2.1 reduces to

0 → Hom(Q, ZN) →^{μ} Aut_{N}(G) →^{ρ} Aut(Q) × Aut(N) → H^{2}(Q, ZN)

Proof: [4], Th. 3.1. Clearly Z1 ; \overline{χ}(Q, ZN) = Hom(Q, ZN) and C_{χ} = Aut(Q) × Aut(N).

Before proceeding, it is instructive to consider the simplest case G = N × Q. The extension class is trivial thus ρ : Aut_{N}(G) → Aut(Q) × Aut(N) is surjective. In fact ρ is split by the usual inclusion Aut(Q) × Aut(N) → Aut_{N}(G). For a more complete discussion of Aut(Q × N) see [11].

In what follows we are interested in G = P, a p-group. As an immediate corollary of 4.1 we have

Corollary 4.2. Let P be a p-group and define P_{1} = P, P_{i+1} = P_{i}/Z(P_{i}). Suppose Aut(Z(P_{i})) is a p-group for each i. Then Aut(P) is a p-group. In particular this hypothesis holds if p = 2 and for each i the summands of P_{i} have distinct exponents.

Proof: The first statement is clear from Corollary 2.3. Similarly the second follows from Proposition 4.5 of [11] which implies each Aut(P_{i}) is a 2-group.

Another particularly tractable case is that of E : N → P → Q, with N an Abelian, characteristic subgroup. Since Hom(Q, N) tends to be relatively large in this case, the quotient Im(ρ) can be significantly smaller than Aut(G). We examine this phenomenon in more detail.

Let {Γ^{n}(P)} be the mod-p lower central series, that is Γ^{0}(P) = P and

Γ^{n}(P) = ⟨(g_{1}, …, g_{s})^{p^{k}} | sp^{k} > n ⟩, n ⩾ 1

where (g_{1}, …, g_{s}) = (g_{1}, (g_{2}, (… (g_{s−1}, g_{s}) …)) is the s-fold iterated commutator. Then Γ^{1}(P) = Φ(P) is the Frattini subgroup and V = P/Γ^{1}(P) is the largest elementary Abelian quotient of P. For some n, Γ^{n}(P) = 1.

Example 4.3. For p an odd prime we consider the group

P = ⟨a, b, c | a^{p^{2}} = b^{p^{2}} = c^{p} = 1, c = (b, a), trivial higher commutators ⟩ We shall determine Aut(P/Γ^{p}(P)) using the extension

E : N = Γ^{1}(P)/Γ^{p}(P) → P/Γ^{p}(P) → Q = P/Γ^{1}(P)

where N = Z/p⟨a^{p}, b^{p}, c⟩ and Q = Z/p⟨\overline{a}, \overline{b}⟩ are elementary Abelian. This extension has trivial twisting and its extension cocycle is easily seen to be [E] = (x ∧ y) ⊗ c ∈ H^{2}(V, Q) = H^{2}(V) ⊗ Z/p⟨z⟩ where x, y are dual to a, b respectively. By inspection

Im(ρ) = ⟨(A, B) ∈ GL_{2}(F_{p}) × GL_{1}(F_{p}) | B = det(A)⟩.

Thus Im(ρ) ≅ GL_{2}(F_{p}). We conclude F_{p}Aut(P) → F_{p}GL_{2}(F_{p}) is surjective with nilpotent kernel. Finally we note Hom(Q, N) = Mat_{3,2}(F_{p}), thus |Hom(Q, N)| = p^{6} and |Aut(P)| = p^{6}|GL_{2}(F_{p})| = p^{7}(p^{2} − 1)(p − 1).

Example 4.4. Let U_{5}(F_{2}) ⩽ GL_{5}(F_{2}) be the unipotent subgroup of upper triangular matrices over F_{2}. These matrices necessarily have ones on the diagonal. Thus U_{5}(F_{2}) is generated by x_{ij} = I_{4} + e_{ij} where 1 ⩽ i < j ⩽ 5 and e_{ij} is the standard elementary matrix with 1 in the ij position and zeros elsewhere. Let P = U_{5}(F_{2})/Γ^{2}U_{5}(F_{2}). Then there is a central extension

E : N = Z/2⟨\overline{x}_{13}, \overline{x}_{24}, \overline{x}_{35}⟩ → P → Q = Z/2⟨\overline{x}_{12}, \overline{x}_{23}, \overline{x}_{34}, \overline{x}_{45}⟩

Since N is the commutator subgroup of P, it is characteristic. The extension cocycle is

[E] = y_{12}y_{23} ⊗ \overline{x}_{13} + y_{23}y_{34} ⊗ \overline{x}_{24} + y_{34}y_{45} ⊗ \overline{x}_{35}

where

H^{∗}(Q, N) = H^{∗}(Q) ⊗ N = Z/2⟨y_{12}, y_{23}, y_{34}, y_{45}⟩ ⊗ ⟨\overline{x}_{13}, \overline{x}_{24}, \overline{x}_{35}⟩

and y_{ij} is dual to \overline{x}_{ij}. Since the twisting is trivial, C_{χ} = Aut(Q) × Aut(N) = GL_{4}(F_{2}) × GL_{3}(F_{2}). Direct calculation shows

(Aut(Q) × Aut(N))_{[E]} = ⟨A, B | A^{4} = B^{2} = 1, BAB = A^{−1}⟩

the dihedral group of order 8 generated by

A = (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \end{matrix}) × (\begin{matrix}0 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{matrix})

B = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}) × (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix})

Moreover |Hom(Q, N)| = 2^{12}, Aut(P) is a 2 group of order 2^{15}.

5. An inductive procedure for determining Aut(P)

Let G = P be a p-group. Since Γ^{i}(P) is characteristic there is an induced homomorphism ρ_{V} : Aut(P) → Aut(V) which factors as

ρ_{V} : Aut(P) → ⋯ → Aut(P/Γ^{i+1}(P)) →^{ρ_{i}} Aut(P/Γ^{i}(P))

→ ⋯ →^{ρ_{2}} Aut(P/Γ^{2}(P)) →^{ρ_{1}} Aut(V)

We shall describe an inductive procedure for lifting elements in the image of this map.

Consider the extensions

E_{i} : Γ^{1}(P)/Γ^{i}(P) → P/Γ^{i}(P) → V, i ⩾ 2
Ẽ_{i} : Γ^{i}(P)/Γ^{i+1}(P) → Γ^{1}(P)/Γ^{i+1}(P) → Γ^{1}(P)/Γ^{i}(P), i ⩾ 2.

where E_{2} and Ẽ_{i} have trivial twisting. In each case the kernel is a characteristic subgroup.

Let σ_{1} ∈ Aut(V). By Theorem 2.1,

σ_{1} ∈ Im{Aut_{Γ^{1}(P)/Γ^{2}(P)}(P/Γ^{2}(P)) → Aut(V)}

if and only if there exist τ_{1} ∈ Aut(Γ^{1}(P)/Γ^{2}(P)) such that (σ_{1}, τ_{1}) fixes

[E_{2}] ∈ H^{2}(V, Γ^{1}(P)/Γ^{2}(P)).

Then there exists σ_{2} ∈ Aut_{Γ^{1}(P)/Γ^{2}(P)}(P/Γ^{2}(P)) lifting σ_{1}. Since

Aut_{Γ^{1}(P)/Γ^{2}(P)}(P/Γ^{2}(P)) = Aut(P/Γ^{2}(P))

this completes the initial step. Now suppose inductively that we have found elements σ_{i} ∈ Aut(P/Γ^{i}(P)), τ_{i−1} ∈ Aut(Γ^{1}(P)/Γ^{i}(P)) such that σ_{i}, τ_{i−1} are lifts of σ_{1}, τ_{i−2}, respectively. We need to find τ_{i} ∈ Aut(Γ^{1}(P)/Γ^{i+1}(P)) such that (σ_{1}, τ_{i}) ∈ C_{χ} fixes

[E_{i+1}] ∈ H2 ; \overline{χ}(V, Z(Γ^{1}(P)/Γ^{i+1}(P))).

Then by Thm 2.1 there exists σ_{i+1} ∈ Aut(P/Γ^{i}(P)) lifting σ_{1}.

To find τ_{i} we apply the same technique to the extension Ẽ_{i} noting that the twisting is trivial so condition (1) is trivially satisfied. Thus we must find τ′_{i} ∈ Aut(Γ^{i}(P)/Γ^{i+1}(P)) such that (τ_{i−1}, τ′_{i}) fixes

[Ẽ_{i}] ∈ H^{2}(Γ^{1}(P)/Γ^{i}(P), Γ^{i}(P)/Γ^{i+1}(P))

since Γ^{i}(P)/Γ^{i+1}(P) is its own center. Then applying Theorem 2.1 again there exists τ_{i} ∈ Aut(Γ^{1}(P)/Γ^{i+1}(P)) lifting τ_{i−1} as desired.

The induction terminates when Γ^{i+1}(P) = 1.

The procedure described in this section is demonstrated in the examples in Section 7.

6. Extensions of elementary Abelian groups

We consider extensions E : N → G → V where N, V are elementary Abelian pgroups with N central. In this case the twisting is trivial, χ = id and C_{χ} = Aut(N)× Aut(V). Our aim is to use Corollary 3.3 to compute |Im(ρ)|. Let n = dim_{F_{p}}(N). Then the extension cocycle [E] ∈ H^{2}(V; N) has the form [E] = (X_{1}, X_{2}, …, X_{n}) where X_{i} ∈ H^{2}(V; F_{p}). We recall that Aut(V) acts diagonally on [E], σ[E] = (σX_{1}, σX_{2}, …, σX_{n}) for σ ∈ Aut(V). Thus the isotropy subgroup

Aut(V)_{[E]} = Aut(V)_{X_{1}} ∩ ⋯ ∩ Aut(V)_{X_{n}} (3)

The action of Aut(N) on [E] is induced from that on N.

6.1. Quadratic Forms

At this point we restrict our attention to the case p = 2. Let m = dim(V) then each X_{i} is a quadratic form in x_{1}, x_{2}, …, x_{m} the generators of H^{∗}(V; F_{2}) = F_{2}[x_{1}, x_{2}, …, x_{m}], |x_{i}| = 1.

We recall some classical facts about quadratic forms Q : V → F_{2}, [3], [6], [5]. The defining property is that the associated form B(x, y) = Q(x + y) + Q(x) + Q(y) is alternate bilinear.

The bilinear radical of B,

bilrad(V, B) := {x ∈ V |B(x, y) = 0, ∀y ∈ V }

As usual, B is called non-degenerate if bilrad(V, B) = 0, i.e. its matrix is nonsingular. The radical of Q,

Rad(V, Q) := {x ∈ bilrad(V, B)|Q(x) = 0}

Q is said to be non-degenerate if Rad(V, Q) = 0.

By a theorem of Dickson [6] (Section 199) a (non-zero) quadratic form over F_{2} in m variables which is not equivalent (by a change of basis) to one in fewer variables must be equivalent to one of the following standard non-degenerate quadratic forms

Φ_{m}^{+} = x_{1}x_{2} + ⋯ + x_{m−1}x_{m}, m even
Φ_{m}^{−} = x_{1}x_{2} + ⋯ + x_{m−3}x_{m−2} + x_{m−1}^{2} + x_{m−1}x_{m} + x_{m}^{2}, m even
Φ_{m} = x_{1}^{2} + x_{2}x_{3} + ⋯ + x_{m−1}x_{m}, m odd

If bilrad(V, B) = 0, then m = 2r is even and one can define the Arf invariant of Q with respect to a symplectic basis {u_{1}, v_{1}, …, u_{k}, v_{k}} by

Arf(Q) = ∑_{i=1}^{k} Q(u_{i})Q(v_{i}) ∈ Z/2

This is invariant of the choice of symplectic basis and determines Q up to equivalence. It is convenient to write Z/2 = ⟨±1⟩ multiplicatively. Then with this notation, W. Browder has shown that Arf(Q) = 1, −1 if and only if Q sends the majority of elements of V to 1, −1 respectively [3]. For m even one finds Arf(Φ_{m}^{+}) = 1, Arf(Φ_{m}^{−}) = −1.

The Arf invariant can be extended to the m odd case (where B is degenerate) as follows. It is easy to see that Φ_{m} sends the same number of elements to 1 and −1 thus one can define Arf(Φ_{m}) = 0. It is clear that Browder’s definition (also known as the “democratic invariant”) is invariant under any basis change. This leads to the following classification theorem.

Theorem 6.1. [3],Theorem III.1.14 A quadratic form Q : V → Z/2 is determined up to equivalence by the triple (dim(V), dim(bilrad(V, B)), Arf(Q))

The action of Aut(N) = GL(n, F_{2}) on an n-tuple (X_{1}, X_{2}, …, X_{n}) of quadratic forms is linear and thus involves the sum of forms. Unfortunately it is impossible, in general, to determine the sum from the Arf invariant. For example if X_{1} = x^{2} and X_{2} = xy + y^{2} then Arf(X_{1} + X_{2}) ≠ Arf(X_{1}) + Arf(X_{2}). However, on a direct sum of vector spaces then it follows easily from the definition that

Arf(X_{1} ⊕ X_{2}) = Arf(X_{1}) + Arf(X_{2}).

For the rest of this subsection we restrict attention to m even and consider the special case [E] = (X_{1}, X_{2}, …, X_{n}) where the X_{i} are in standard form. We write X_{i} = X or Y depending on whether the Arf invariant is 1 or −1. In the following theorem we use ∼ to denote conjugacy.

Theorem 6.2. Suppose m = 2r and [E] = (X_{1}, X_{2}, …, X_{n}) with X_{1} = ⋯ = X_{k} = X, X_{k+1} = ⋯ = X_{n} = Y. Then Ω([E]) = [E] and |Im(ρ)| = |Aut(V)_{[E]}| · |Aut(N)_{[E]}|. Furthermore

1) If k = n then

Aut(V)_{[E]} = O_{m}^{+}(F_{2})

the orthogonal group of order 2(2^{r} − 1) ∏_{i=1}^{r−1}(2^{2i} − 1)2^{2i} of matrices preserving the form X.

Aut(N)_{[E]} ∼ (\begin{matrix}1 & 0 \\ ∗ & GL_{n−1}(F_{2}) \end{matrix})

of order 2^{n−1} ∏_{i=1}^{n−1}(2^{i} − 1)2^{i−1}. Im(ρ) is a 2-group if and only if m, n < 3.

2) If k = 0 then

Aut(V)_{[E]} = O_{m}^{−}(F_{2})

the orthogonal group of order 2(2^{n} + 1) ∏_{i=1}^{n−1}(2^{2i} − 1)2^{2i} of matrices preserving the form Y and

Aut(N)_{[E]} ∼ (\begin{matrix}1 & 0 \\ ∗ & GL_{n−1}(F_{2}) \end{matrix}) .

Im(ρ) is a 2-group if and only if m < 2 and n < 3.

3) If 1 < k < n then

Aut(V)_{[E]} = O_{m}^{+}(F_{2}) ∩ O_{m}^{−}(F_{2})

and

Aut(N)_{[E]} ∼ (\begin{matrix}I_{2} & 0 \\ ∗ & GL_{n−2}(F_{2}) \end{matrix})

of order 2^{2(n−2)} ∏_{i=1}^{n−2}(2^{i} − 1)2^{i−1}. Im(ρ) is a 2-group if and only if m, n < 4.

Proof: The calculation of Aut(V)_{[E]} follows from (3). The intersection orbit group Ω([E]) = (Aut(V)·[E])∩(Aut(N)·[E]) = {[E]}. To see this suppose σ[E] = [E]τ^{−1} differs from [E] = (X_{1}, X_{2}, …, X_{n}) in the i-th coordinate X_{i} = X, say. Now σX = ([E]τ^{−}1)_{i} = aX + bY, a, b ∈ F_{2}. Thus σX = X + Y since σX ≠ X and Y has Arf invariant −1. However σX = X + Y = x_{m−1}^{2} + x_{m}^{2} = (x_{m−1} + x_{m})^{2} contradicting the fact that X is not equivalent to a quadratic form in fewer than m variables. Similarly if X_{i} = Y.

Since X and Y are linearly independent polynomials, (X, …, X, Y, …, Y) is equivalent to (X, Y, 0, …, 0) by a change of basis. Thus the descriptions of Aut(N)_{[E]} follow immediately.

6.2. Non-standard Forms

We now turn to the more general case where the forms X_{i} of [E] = (X_{1}, X_{2}, …, X_{n}) are not in standard form. One can still determine |Im(ρ)|; we illustrate this by considering the case n = 2, m = 3. Thus we consider pairs of quadratic forms (X, Y) in variables x, y, z each equivalent to (but not necessarily equal to) one of the standard four forms:

x^{2} + yz, xy, x^{2} + xy + y^{2}, x^{2}.

The respective isotropy groups, as subgroups of GL_{3}(F_{2}), are

Aut(V)_{x^{2}+yz} = O_{3}(F_{2}) ≅ GL_{2}(F_{2})

non-Abelian of order 6;

Aut(V)_{xy} = (\begin{matrix}Σ_{2} & 0 \\ ∗ & 1 \end{matrix})

elementary Abelian of rank 3;

Aut(V)_{x^{2}+xy+y^{2}} = (\begin{matrix}GL_{2} & 0 \\ ∗ & 1 \end{matrix})

of order 24;

Aut(V)_{x^{2}} = (\begin{matrix}1 & 0 \\ ∗ & GL_{2} \end{matrix})

of order 24.

6.2.1. Simultaneous Equivalence First we consider the case where X and Y are simultaneously equivalent to a standard form, i.e., there is an invertible linear transformation A of V such that A^{−1}XA and A^{−1}Y A are each in standard form.

Case 1, X = Y:

Then Aut(V)_{(X,X)} = Aut(V)_{X}. It is also easy to see Aut(N)_{(X,X)} = Σ_{2} and the intersection of orbits Ω([E]) = {(X, X)} in this case.

a) X is equivalent to xy : As above the isotropy subgroup Aut(V)_{(X,X)} is elementary Abelian of rank 3. The intersection of the orbits is just (X, X) so the order of Im(ρ) = 8 · 2 · 1 = 16.

b) X is equivalent to x^{2} + xy + y^{2} : Aut(V)_{(X,X)} is of order 24. Thus |Im(ρ)| = 24 · 2 · 1 = 48.

c) X is equivalent to x^{2}+yz : Aut(V)_{(X,X)} = O_{3}(F_{2}). Thus |Im(ρ)| = 6·2·1 = 12.

d) X is equivalent to x^{2}: As above Aut(V)_{(X,X)} is of order 24. Thus |Im(ρ)| = 24 · 2 · 1 = 48.

Case 2, X ≠ Y:

In this case Aut(N)_{(X,Y)} = 1. Further Aut(V)_{(X,Y)} = Aut(V)_{X} ∩ Aut(V)_{Y} = Aut(V)_{(Y,X)}. There are several possibilities; we give only four in detail since the rest follow the same general pattern. By considering the Arf invariant we see that Ω([E]) = 1 except in the third example.

1) (X, Y) equivalent to (x^{2} + yz, x^{2} + xy + y^{2}): Then Aut(V)_{X} ∩ Aut(V)_{Y} is

O_{3}(F_{2}) ∩ (\begin{matrix}GL_{2} & 0 \\ ∗ & 1 \end{matrix}) = Z/2⟨ (\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}) ⟩.

The intersection of orbits is {(X, Y)} hence |Im(ρ)| = 2 · 1 · 1 = 2.

2) (X, Y) equivalent to (xy, x^{2} + xy + y^{2}): Then Aut(V)_{X} ∩ Aut(V)_{Y} is

(\begin{matrix}Σ_{2} & 0 \\ ∗ & 1 \end{matrix}) ∩ (\begin{matrix}GL_{2} & 0 \\ ∗ & 1 \end{matrix}) = (\begin{matrix}Σ_{2} & 0 \\ ∗ & 1 \end{matrix})

i.e., dihedral of order 8. The intersection of orbits is {(X, Y)} thus |Im(ρ)| = 8·1·1 = 8.

Similarly for (xy, y^{2} + yz + z^{2}) and (xz, x^{2} + xy + y^{2}).

3) (X, Y) equivalent to (xy, x^{2} + yz): Then Aut(V)_{X} ∩ Aut(V)_{Y} is trivial and Ω([E]) = {(X, Y), (X + Y, Y)} = Z/2. Thus |Im(ρ)| = 2.

4) (X, Y) equivalent to (xy, yz): Then Aut(V)_{X} ∩ Aut(V)_{Y} is trivial. Direct calculation shows the intersection of the orbits has order 6. Thus |Im(ρ)| = 1·1·6 = 6.

Similarly for (xz, yz), (xz, xy), and (yz, xz).

6.2.2. Non-simultaneous Equivalence By this we mean X and Y are not simultaneously equivalent to a pair of standard forms. Since X ≠ Y, Aut(N)_{(X,Y)} = 1 in all cases.

To illustrate this phenomena the following table gives a complete computation of |Im(ρ)| in case X and Y are equivalent (non-simultaneously) to x^{2} + yz. Then X may be assumed to be x^{2}+yz and we list only the relevant Y ’s and the corresponding values of |Im(ρ)| = 2, 3, 4.

|Im(ρ)|
2  3  4
x^{2} + xz + yz  xy + xz + y^{2}  x^{2} + yz + z^{2}
x^{2} + xy + xz + z^{2} x^{2} + xy + xz + y^{2} + yz xy + xz + y^{2} + yz + z^{2}
x^{2} + xz + yz + z^{2}  xz + y^{2} + yz  xy + xz + yz
x^{2} + xy + y^{2} + yz  xy + xz + z^{2}  xz + y^{2} + z^{2}
x^{2} + xy + yz  x^{2} + xy + y^{2} + z^{2}  x^{2} + y^{2} + yz
x^{2} + xy + xz + y^{2}  x^{2} + xz + y^{2} + z^{2}  xy + z^{2}
x^{2} + xy + z^{2}  xy + y^{2} + z^{2}
xz + y^{2} + yz + z^{2}  x^{2} + y^{2} + yz + z^{2}
x^{2} + xz + y^{2}  xz + y^{2}
x^{2} + xy + xz + yz + z^{2}
xy + y^{2} + yz + z^{2}
xy + yz + z^{2}

In more detail, if |Im(ρ)| = 2, then Aut(V)_{(X,Y)} = 1 and Ω([E]) = {(X, Y), (Y, X)} = Z/2 If |Im(ρ)| = 3, then |Aut(V)_{(X,Y)}| = 1 and Ω([E]) = {(X, Y), (X+Y, X), (Y, X+ Y)} = Z/3. If |Im(ρ)| = 4 then |Aut(V)_{(X,Y)}| = 2 and Ω([E]) = {(X, Y), (Y, X)} = Z/2.

As a final example we consider

X = xy + y^{2}, Y = x^{2} + y^{2} + yz + z^{2}. To analyze Aut(V)_{(X,Y)} we separately reduce X, Y to standard forms f_{1}, f_{2} respectively

σ_{1}X = f_{1}, σ_{2}Y = f_{2}

with σ_{i} ∈ Aut(V). Let σ = σ_{2}σ_{1}^{−1}, then σ^{−1}f_{2} = σ_{1}Y hence

Aut(V)_{(f_{1},σ_{1}Y)} = Aut(V)_{f_{1}} ∩ σ^{−1}[Aut(V)_{f_{2}}]σ

This determines Aut(V)_{(X,Y)} up to conjugacy since Aut(V)_{(X,Y)} = σ_{1}^{−1}[Aut(V)_{(f_{1},σ_{1}Y)}]σ_{1}. In this example f_{1} = xy, f_{2} = x^{2} + yz with σ_{1} : y ↦ x + y, σ_{2} = x ↦ x + y + z. Then

Aut(V)_{(f_{1},σ_{1}Y)} = Aut_{xy} ∩ σ^{−1}[Aut(V)_{x^{2}+yz}]σ = Z/2⟨ (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \end{matrix}) ⟩

Further Ω([E]) = {(X, Y), (X, X + Y)} = Z/2 hence |Im(ρ)| = 2 · 1 · 2 = 4.

7. Applications

1. Let P denote the extraspecial group of order |P | = p^{2n+1} and exponent p > 2 defined by the central extension

E : Z → P → V

where Z = Φ(P) = Z/p, V = P/Φ(P) = (Z/p)^{2n}. The twisting χ is trivial thus C_{χ} = Aut(V) × Aut(Z). Complete information about the automorphism group of P as well as all other extensions of elementary Abelian p-groups by Z/p is known [15],[7]. Here we apply our results to obtain a quick derivation of Im{Aut(P) →^{ρ} Aut(V) × Aut(Z)}.

P is generated by elements x_{1}, x_{2}, …, x_{2n}, ζ of order p satisfying

[ζ, x_{i}] = 1
[x_{2i−1}, x_{2i}] = ζ
[x_{2i−1}, x_{j}] = 1, j ≠ 2i
[x_{2i}, x_{j}] = 1, j ≠ 2i − 1

Thus Φ(P) = Z(P) = ⟨ζ⟩ = Z/p and V = ⟨\overline{x}_{1}, \overline{x}_{2}, …, \overline{x}_{2n}⟩. It is immediate from these relations that the extension cocycle is

[E] = B ⊗ ζ

where B = y_{1}y_{2} + ⋯ + y_{2n−1}y_{2n}, y_{i} ∈ H^{1}(V) is dual to \overline{x}_{i}. Since p is an odd prime, y_{i}y_{j} = −y_{j}y_{i}. Thus B is exactly the skew-symmetric form for the symplectic group Sp(n, F_{p}). We shall need a slightly more general version GSp(2n, F_{p}), [5], the transformations which fix B up to a scalar. It is easy to see that GSp(2n, F_{p}) = ⟨γ⟩ ⋊ Sp(2n, F_{p}) where γ is the linear transformation

\overline{x}_{2i−1} ↦ k\overline{x}_{2i−1}, \overline{x}_{2i} ↦ \overline{x}_{2i}

and k is a generator of F_{p}^{∗}.

Now considering (σ, τ) ∈ Aut(Z) × Aut(V) acting on E we find

(σ, τ)(E) = [∑_{i=1}^{n}(σ)(y_{2i−1})σ(y_{2i})] ⊗ τ(ζ)

Thus (σ, τ)(E) = E if and only if σ ∈ GSp(2n, F_{p}) and τ is multiplication by det(σ)^{−1}. We conclude Im{Aut(P) → Aut(V) × Aut(Z)} ≅ GSp(2n, F_{p}).

2. Let W (n) be the universal W-group [1] on n generators defined as the central extension

1 → N = Φ(W (n)) → W (n) → Q = (Z/2)^{n} → 1

where N = (Z/2)^{n+\binom{n}{2}}. These groups arise, for instance, in the study of the cohomology of Galois groups. The extension class [E] = (X_{1}, X_{2}, …, X_{n+\binom{n}{2}}) where the

{X_{i}} form an ordered basis for H^{2}(Q) = S^{2}(Q^{∗}), the second symmetric power of the dual of Q. Any order will do. The twisting is trivial, hence C_{χ} = GL(Q) × GL(N). Now GL(Q) induces linear isomorphisms on H^{∗}(Q) which are determined by their values on squares in H^{2}(Q). Thus given σ ∈ GL(Q) we can find a τ ∈ GL(N) such that (σ, τ)([E]) = (σ[E])τ^{−1} = [E]. Thus we see that Im(ρ) ⩽ GL(Q) × GL(N) by an injection which projects to an isomorphism on the first factor, i.e. Im(ρ) ≅ GL(Q).

3. We consider Aut(P) for the unipotent group P = U_{4}(F_{2}) ⊂ GL_{4}(F_{2}) of upper triangular matrices over the finite field F_{2}. These matrices necessarily have ones on the diagonal. Thus P is generated by {x_{12}, x_{23}, x_{34}}. Then Γ^{3}(P) = 1, Γ^{2}(P) = Z/2⟨x_{14}⟩ = ZP, and Γ^{1}(P) = Z/2⟨x_{13}, x_{24}, x_{14}⟩. The corresponding extensions have

E_{2} : Γ^{1}(P)/Γ^{2}(P) →^{i_{2}} P/Γ^{2}(P) → V

where V = P/Γ^{1}(P) = Z/2⟨\overline{x}_{12}, \overline{x}_{23}, \overline{x}_{34}⟩, Γ^{1}(P)/Γ^{2}(P) = Z/2⟨\overline{x}_{13}, \overline{x}_{24}⟩ and

E_{3} : Γ^{1}(P) →^{i_{3}} P → V

The twisting χ is trivial for E_{2} but not for E_{3}. In this application we shall not need the auxiliary extensions [Ẽ_{i}].

Claim: Im{Aut(P) → Aut(V)} ≅ Σ_{3}

Proof: (Sketch) From the commutator relations [x_{ij}, x_{jk}] = x_{ik} for i < j < k, one sees that the extension class [E_{2}] ∈ H^{2}(V; Γ^{1}(P)/Γ^{2}(P)) is xy ⊗ \overline{x}_{13} + yz ⊗ \overline{x}_{24} where

H^{∗}(V, Γ^{1}(P)/Γ^{2}(P)) = H^{∗}(V) ⊗ Z/2⟨x_{13}, x_{24}⟩
= Z/2[x, y, z] ⊗ Z/2⟨x_{13}, x_{24}⟩

Here x, y, z ∈ H^{1}(V) are classes dual to \overline{x}_{12}, \overline{x}_{23}, \overline{x}_{34} respectively. Direct calculation shows the [E_{2}] is fixed the subgroup generated by the involution defined by

σ : \overline{x}_{12} ↦ \overline{x}_{34}, \overline{x}_{23} ↦ \overline{x}_{23}, τ : x_{13} ↦ x_{24}

and the map of order three

σ′ : \overline{x}_{12} ↦ \overline{x}_{12} + \overline{x}_{34}, \overline{x}_{23} ↦ \overline{x}_{23}, \overline{x}_{34} ↦ \overline{x}_{12},
τ′ : x_{13} ↦ x_{24}, x_{24} ↦ x_{13} + x_{24}

One finds ⟨(σ, τ), (σ′, τ′)⟩ ≅ Σ_{3}.

Turning to the extension E_{3}, we can extend τ, τ′ to Γ^{1}(P) by letting them act identically on x_{14}. Then a simple calculation shows ⟨(σ, τ), (σ′, τ′)⟩ ⩽ C_{χ}. We note that [E_{3}] = [E_{2}] considered as an element of H2 ; \overline{χ}(V; Γ^{1}(P)). This follows from the definition of the cocycle

f : V × V → Γ^{1}(P)/Γ^{2}(P)

for [E_{2}]. Recall that given a set theoretic section s : V → P/Γ^{2}(P) then

s(a)s(b) = i_{2}(f(a, b))s(ab)

If we use one of the sections not involving the center, then it lifts to a section s̃: V → P and the corresponding cocycle f̃: V ×V → Γ^{1}(P) is a lift of f. (For example using the ordered basis (\overline{x}_{12}, \overline{x}_{23}, \overline{x}_{34}) for V let s̃(\overline{x}_{12}) = x_{12}, s̃(\overline{x}_{23}) = x_{23}, s̃(\overline{x}_{34}) = x_{34}. For products w ∈ V let s̃(w) = w′ where w′ is ordered lexicographically. Thus if w = \overline{x}_{34}\overline{x}_{23}\overline{x}_{12} then w′ = x_{12}x_{23}x_{34}. Hence

s̃(\overline{x}_{34}\overline{x}_{23})s̃(\overline{x}_{12}) = i_{3}(f̃(\overline{x}_{34}\overline{x}_{23}, \overline{x}_{12}))s̃(\overline{x}_{34}\overline{x}_{23}\overline{x}_{12})

implies f̃(\overline{x}_{34}\overline{x}_{23}, \overline{x}_{12}) = x_{13}x_{24} In general we observe that no two-fold products among the elements of s̃(V) involve the center i.e., two-fold products among the elements x_{12}, x_{23}, x_{34} are on the first and second diagonals, not in the center.)

This means (σ, τ), (σ′, τ′) fix the extension class [E_{3}] and hence define automorphisms of P not just P/Γ^{1}(P).

4. Let p be an odd prime and let

P = ⟨a, b, c, d | a^{p} = b^{p} = c^{p} = d^{p} = 1, c = (b, a), d = (c, a), other commutators trivial⟩

We shall study Aut(P) for p = 5, the smallest prime for which P is regular. We consider the extensions of V = P/Γ^{1}(P) = ⟨\overline{a}, \overline{b}⟩

[E_{2}] : N_{2} := Γ^{1}(P)/Γ^{2}(P) → P/Γ^{2}(P) → V
[E_{3}] : N_{3} := Γ^{1}(P) → P → V

Since Γ^{3}(P) = 1, we also have

[Ẽ_{2}] : Ñ_{3} := Γ^{2}(P) → Γ^{1}(P) → N_{2}

Each N is Abelian:

N_{2} = ⟨\overline{c}⟩, N_{3} = ⟨c, d⟩, Ñ_{3} = ⟨d⟩

Noting that E_{3} has non-trivial twisting we apply the algorithm of Section 4 to study Aut(P).

Claim: The semi-simple quotient of F_{p}(Aut(P)) is F_{p}(Z/4)^{2}.

Proof: (sketch)

Extension E_{2} has trivial twisting and is quite similar to that of Example 1. Using the same notation and arguing analogously we can determine the extension class [E_{2}] = xy ⊗ z ∈ H^{2}(V, N_{2}). Hence

C_{[E_{2}]} = ⟨(A, B) ∈ GL_{2}(F_{p}) × GL_{1}(F_{p}) | B = det(A)⟩

Extension Ẽ_{2} splits, Γ^{1}(P) = Ñ_{2} × N_{2} = ⟨c, d⟩. Thus every τ ∈ Aut(N_{2}) lifts to Aut(Γ^{1}(P)) and we proceed to study extension E_{3}.

First we determine

C_{χ} = ⟨(σ, τ) ∈ Aut(V) × Aut(N_{3}) | χσ = c\overline{τ} χ⟩

The action of V on N_{2} is given by χ(\overline{x}) : c ↦ c + d, d ↦ d, χ(\overline{y}) = id. Hence ker(χ) = ⟨\overline{y}⟩. Since σ : ker(χ) → ker(χ), it must have the form

σ = (\begin{matrix}k & 0 \\ m & n \end{matrix}) ∈ GL_{2}(F_{p})

Now solving χσ = c\overline{τ} χ where

τ = (\begin{matrix}s & t \\ u & v \end{matrix}) ∈ GL_{2}(F_{p})

we find

(\begin{matrix}1 & 0 \\ k & 1 \end{matrix}) = (\begin{matrix}1 + tv/Δ & t^{2}/Δ \\ v^{2}/Δ & 1 − tv/Δ \end{matrix})

where Δ = det(τ). Thus t = 0 and k = v/s. Hence

C_{χ} = \{(\begin{matrix}k & 0 \\ m & n \end{matrix}) × (\begin{matrix}s & 0 \\ u & v \end{matrix}) | k = v/s\}

To determine Im(ρ) = C_{(}χ)_{[E_{3}]} it is easier in this case to observe that |Aut(P)| = 4^{2}5^{5} (using Magma) and argue directly instead of using the extension class. First ker(ρ) contains a subgroup of order 5^{3} generated by the inner automorphisms of order 5, {c_{b}, c_{c}} ⩽ ker(ρ) together with the automorphism ϕ : ϕ(b) = bd^{−1}, ϕ = id on a, c, d. Next we show that Im(ρ) contains a subgroup of order 4^{2}5^{2} which must equal Im(ρ).

To complete the proof of the claim we note that the subgroup of order 5^{2} is normal.

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John Martino martino@wmich.edu

Department of Mathematics Western Michigan University Kalamazoo, MI 49008 U.S.A.

Stewart Priddy priddy@math.northwestern.edu

Department of Matheamtics Northwestern University Evanston, IL 60208 U.S.A.