Publ. RIMS, Kyoto Univ. (),

The Frattini module and p′-automorphisms of free pro-p groups.

By

Darren Semmen ^{∗}

Abstract

If a non-trivial subgroup A of the group of continuous automorphisms of a noncyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points Fix_{F} (A) has infinite rank.

The semi-direct product F >◁A is the universal p-Frattini cover of a finite group G, and so is the projective limit of a sequence of finite groups starting with G, each a canonical group extension of its predecessor by the Frattini module. Examining appearances of the trivial simple module 11 1 in the Frattini module’s Jordan-Hölder series arose in investigations ([FK97], [BaFr02] and [Sem02]) of modular towers. The number of these appearances prevents Fix_{F} (A) from having finite rank.

For any group A of automorphisms of a group Γ, the set of fixed points Fix_{Γ}(A) := {g ∈ Γ | α(g) = g, ∀α ∈ A} of Γ under the action of A is a subgroup of Γ. Nielsen [N21] and, for the infinite rank case, Schreier [Schr27] showed that any subgroup of a free discrete group will be free. Tate (cf. [Ser02, I.§4.2, Cor. 3a]) extended this to free pro-p groups. In light of this, it is natural to ask for a free group F, what is the rank of Fix_{F} (A)?

When F is a free discrete group and A is finite, Dyer and Scott [DS75] demonstrated that Fix_{F} (A) is a free factor of F, i.e. F is a free product of Fix_{F} (A) and another free subgroup of F, thus bounding the rank of Fix_{F} (A) by that of F itself. That this bound would hold for A that are merely finitely generated was a conjecture attributed to Scott; this was proven first by Gersten [Ge87] and later, independently, by Bestvina and Handel [BH92] in a program analogizing, to outer automorphisms of free groups, Thurston’s classification of mapping classes.

Communicated by

2000 Mathematics Subject Classification(s): 14G32, 20C20, 20D25, 20E05, 20E18, 20F14,

20F28.

Supported by RIMS and Michael D. Fried, October 26 - November 1, 2001.

∗

University of California, Irvine, Irvine, CA 92697-3875, USA

E-mail address: dsemmen@math.uci.edu

But when F is a free pro-p group, this depends on whether the order of A is divisible by p. When A is a finite p-group, Fix_{F} (A) will again be a free factor of F, first shown by Scheiderer [Sche99] for F having finite rank, and extended by Herfort, Ribes, and Zalesskii [HRZ99] to the general case. It is not yet known whether the rank of Fix_{F} (A) will be bounded by that of F when A is an arbitrary (even finitely generated) pro-p group. Contrarily, Herfort and Ribes [HR90] showed much earlier that if F is non-cyclic and α is a nontrivial continuous automorphism of F of finite order, not divisible by p, then Fix_{F} (⟨α⟩) has infinite rank; their proof relies on combinatorial group theory for free pro-p groups and Thompson’s theorem on the nilpotency of finite groups with fixed-point-free automorphisms of prime order.

Using instead the construction of a free pro-p group by taking the projective limit of a canonical sequence of finite p-groups, and the modular representation theory attached to this sequence, we generalize the result of Herfort and Ribes to any non-trivial finite group A of automorphisms having order prime to p, not merely the cyclic case ⟨α⟩.

A note on reading this paper. The first four sections of this paper consist of background material, and state results largely without proof. For results given no explicit reference, the following sources may be consulted. Fried and Jarden [FJ86, Chapters 1, 15, & 20] provide all the coverage we need on profinite groups, and on the universal Frattini cover (§2) as well. For §2 and the first half of §3, also visit Fried’s introduction to modular towers [Fr95, Part II, p.126-136]. Benson has written a dense primer [Be98, Chapter 1] on modular representation theory, which can help with §3 and the sometimes folkloric contents of §4.

§1. Free profinite groups.

A profinite group is a projective (inverse) limit of finite groups, regarded as topological groups with the discrete topology. A morphism in the category of profinite groups is a continuous group homomorphism. One important lemma [FJ86, Lem. 1.2] is that the projective limit of any surjective system (i.e. an inverse system all of whose maps are surjective) of finite sets will surject onto every set in the system.

A profinite group F is free on a set S converging to 1 if and only if it satisfies the following three conditions. First, S ⊆ F must converge to 1 in F, i.e. only a finite number of elements of S lie outside any open subgroup of F.

The Frattini module and p′-automorphisms of free pro-p groups. 3

Second, S must generate F. Third, given any map taking S into a profinite group G such that the image converges to 1 in G, there exists a unique extension of said map to a morphism from F to G. The rank of F is the cardinality of S.

Projective groups are very close to free groups: a profinite group is projective if and only if it is a closed subgroup of a free profinite group. (Hence, closed subgroups of projective groups are also projective.) The property of being projective is categorical. An object X is projective if, whenever there is a morphism φ from X to an object A and an epimorphism ϕ from another object B to A, there exists a morphism φ̂ : X → B such that ϕ ∘ φ̂ = φ. Note that, in the categories of groups and modules, a morphism ϕ is surjective if and only if it is epic: a morphism ϕ : B → A is epic if, given any morphisms ψ_{i} : A → Y for i = 1, 2 with ψ_{1} ∘ ϕ = ψ_{2} ∘ ϕ, ψ_{1} must equal ψ_{2}.

For a rational prime p, a pro-p group is merely a profinite group all of whose finite quotients (by closed normal subgroups) are p-groups. Pro-p groups form a subcategory with the property [FJ86, Prop. 20.37] that all of the projective objects are free with respect to the subcategory, in the sense that the G’s in the above definition of “free” must all be pro-p groups.

Finally, the Schreier formula for free groups holds whether the free groups are discrete, profinite or pro-p:

Theorem 1.1 (Nielsen-Schreier [FJ86, Prop. 15.27]). If a subgroup H of a free group F has finite index, it is free; furthermore, if F has finite rank r, the rank of H is 1 + (r − 1)(G : H).

§2. The universal Frattini cover.

The Frattini subgroup Φ(G) of a profinite group G is the intersection of all maximal proper closed subgroups of G. Note the analogy to the Jacobson radical of an algebra. The Frattini subgroup of a (pro)finite group is also (pro)nilpotent, a consequence of the Frattini argument from whence it gets its name: given a normal subgroup K of G and a p-Sylow P of K, G = N_{G}(P)·K, where N_{G}(P) is the normalizer of P in G.

The Frattini subgroup of a pro-p group has another characterization [FJ86, Lem. 20.36]: it is the closed normal subgroup generated by the commutators and the p^{th}-powers. Put another way, G/Φ(G) is the maximal elementary abelian quotient of the pro-p group G. The Frattini series is just the descending sequence of iterations Φ^{n+1}(G) = Φ(Φ^{n}(G)). This series forms a neighborhood basis of 1 in a pro-p group, i.e. the intersection of all of the terms is trivial.

We can also view “Frattinity” categorically. We say an epimorphism φ : X ↠ A is a Frattini cover if the kernel is in the Frattini subgroup of X.

Equivalently, given any epimorphism ϕ : B ↠ A and any morphism ψ : B → X such that φ ∘ ψ = ϕ, the morphism ψ must be surjective. Note that, in the category of epimorphisms to A, an object which has the property that all morphisms to the object are epic will be a Frattini cover of A. We might call this a Frattini object. The object X is also commonly referred to as the “Frattini cover” of A; we shall do so as well.

Every profinite group G has a universal Frattini cover G̃, a Frattini cover which is projective as a profinite group. (Simply find an epimorphism φ from a free profinite group onto your given group G — the universal Frattini cover will be a minimal closed subgroup H of the free group such that φ(H) = G.) Since the kernel K of our epimorphism from G̃ to G is pronilpotent, i.e. a direct product of its p-Sylows (maximal pro-p subgroups), G̃ will be the fibre product over G of the universal p-Frattini covers _{p}G̃, where _{p}G̃ is just the quotient of G̃ by the maximal closed subgroup of K having no p-group quotient. The universal p-Frattini cover is also characterized by being the unique p-projective Frattini cover of G, p-projective meaning projective only with respect to covers (epimorphisms) with pro-p group kernel; being p-projective is equivalent [FJ86, proof of Prop. 20.47] to having free p-Sylows.

For a finite group G, the universal p-Frattini cover _{p}G̃ ↠^{φ} G will be the projective limit of the finite quotients produced by the Frattini series of the kernel ker_{0} = ker(φ). Inductively define ker_{n+1} = Φ(ker_{n}) and G_{n} = _{p}G̃/ ker_{n}. Since each ker_{n} is a pro-p group, the quotient M_{n} = ker_{n} / ker_{n+1} will be an elementary abelian p-group and, in fact, an 𝔽_{p}G_{n}-module, with the action of an element of G_{n} induced by conjugation, after lifting to G_{n+1}: g · m = ĝmĝ^{−1} for any ĝ such that g = ĝ · ker_{n} / ker_{n+1}.

§3. The Frattini module.

Assume now that G is finite. In the preceeding discussion, we produced a canonical sequence of finite groups whose projective limit was the universal p-Frattini cover _{p}G̃, but only by taking quotients of _{p}G̃. This approach depends on knowledge of _{p}G̃, currently a mysterious object. Fortunately, we may inductively construct G_{n+1} using the modular representations of G_{n}.

For 𝔽_{p}G-modules, projectivity has the same categorical definition given in §1. Several properties are analogous to those given earlier for profinite groups. An 𝔽_{p}G-module is projective if and only if it is a direct summand of a free module. It is also projective if and only if [Be98, Cor. 3.6.10] its restriction to a p-Sylow P of G is a free 𝔽_{p}P -module. We will denote the restriction of an 𝔽_{p}G-module M to a subgroup X of G by M↓_{𝔽_{p}X}.

The projective cover ℙ_{𝔽_{p}G}(M) of a finitely generated 𝔽_{p}G-module M is the minimal projective 𝔽_{p}G-module which has an epimorphism ϕ : ℙ_{𝔽_{p}G}(M) ↠ M; the kernel is denoted ΩM — Ω is known as the Heller operator. An 𝔽_{p}G-module S is called simple if it has no proper non-trivial 𝔽_{p}G-submodules. There is always a 1-dimensional simple 𝔽_{p}G-module 11 1_{𝔽_{p}G} having trivial Gaction; when there will be no ambiguity, the subscript identifying the group ring may be omitted. Gaschütz [Ga54] produced the Frattini module by iterating the Heller operator twice on 11 1_{𝔽_{p}G}:

Theorem 3.1 (Gaschütz [Fr95, Lem. 2.3, p.128]).

As 𝔽_{p}G-modules, M_{0} ≃ Ω^{2}11 1_{𝔽_{p}G} = Ω(Ω11 1_{𝔽_{p}G}).

The Heller operator is the dimension-shift operator on group cohomology [Be98, Prop. 2.5.7], so H^{2}(G, M_{0}) will be 1-dimensional [Fr95, Prop. 2.7, p.132] and there will be only one non-split extension of G by M_{0}, up to isomorphism of groups. This will be G_{1}.

Since _{p}G̃ is also the universal p-Frattini cover of G_{1}, we may use induction to see that M_{n} ≃ Ω^{2}11 1_{𝔽_{p}G_{n}} and G_{n+1} will be the unique non-split extension of G_{n} by M_{n}.

A block of 𝔽_{p}G is an indecomposable two-sided ideal direct summand of the ring 𝔽_{p}G. Every indecomposable (i.e. having no proper non-trivial direct summands) 𝔽_{p}G-module M is contained in some block B; this means that B′ · M = 0 for every block B′ ≠ B. The principal block is the one containing 11 1_{𝔽_{p}G} The kernel of a block is simply the set of elements of G that act trivially on all modules contained in the block, i.e. the kernel of the composition G ↪ 𝔽_{p}G ↠ B mapping G into the units of the ring B.

We now record a few results on the Frattini module. The standard notation for the maximal normal p′-subgroup (i.e. having order prime to p) of G is O_{p′}(G). Using Brauer’s identification of O_{p′}(G) as the kernel of the principal block, Griess and Schmid [GS78] proved that O_{p′}(G) was exactly the kernel of the action of G on M_{0} whenever M_{0} had dimension greater than one. They also identified precisely when the latter happens:

Theorem 3.2 (Griess-Schmid [GS78, Cor. 3, p.264]). The dimension of M_{0} over 𝔽_{p} is one if and only if G is p-supersolvable with cyclic p-Sylows.

A group G is p-supersolvable if and only if G/O_{p′}(G) has a normal p-Sylow such that the quotient is abelian of exponent dividing p − 1; this is quite restrictive.

We denote by #_{S}(M) the number of appearances of a simple 𝔽_{p}G-module S in a Jordan-Hölder series of a given 𝔽_{p}G-module M. Define the density ρ_{S}(M) of S in M to be #_{S}(M)/dim_{𝔽_{p}}(M).

Theorem 3.3 (Density Theorem [Sem02]). If dim_{𝔽_{p}}(M_{0}) ≠ 1 then lim_{n→∞}ρ_{S}(M_{n}) = ρ_{S}(𝔽_{p}G/O_{p′}(G)), for any simple 𝔽_{p}G-module S.

The converse will also hold unless G/O_{p′}(G) is a cyclic p-group.

§4. Groups with normal p-Sylow.

Throughout this section, G will be a finite group with normal p-Sylow P and complement A, i.e. G ≃ P >◁A, the semi-direct product.

In this case, the universal p-Frattini cover has a simpler description. Suppose r is the minimal number of generators of P and F is a free pro-p group having rank r and a given epimorphism ϕ : F ↠ P. Then A has an embedding into the continuous automorphisms of F such that its action will stabilize ker(ϕ) and its action on the quotient F/ ker(ϕ) will correspond to its action on P via the canonical isomorphism. The universal p-Frattini cover of G will be the semi-direct product F >◁A defined by this action, cf. [BaFr02, Rem. 5.2] and [R85].

Example. [Fr95, §II.A, p.126] If G = D_{p} is the dihedral group of order 2p, then _{p}G̃ will be the semi-direct product ℤ_{p}>◁C_{2}, where conjugation by the non-trivial element of the group C_{2} of order 2 inverts the elements of the padic integers ℤ_{p}; the canonical quotients G_{n} will be the dihedral groups D_{p^{n+1}} of order 2p^{n+1}. (These quotient groups appear as the Galois groups for covers of the punctured projective sphere in the Hurwitz space construction of the sequence of modular curves X_{0}(p^{n}); replacing D_{p^{n+1}} by G_{n} leads us to Fried’s [Fr95, §III.C, p.144] modular towers when G is centerless and p-perfect. The fact [FK97, Lem. 3.2, p.167] that obstruction of components of the Hurwitz spaces in modular towers can only arise from appearances of 11 1_{𝔽_{p}G} in a Jordan-Hölder series of M_{0} motivated the examination of ρ11 1(M_{n}).) The modules M_{n} in this case are 1-dimensional over 𝔽_{p}, with the p-Sylow of D_{p^{n+1}} acting trivially and the reflection in C_{2} acting via mulitplication by −1.

When P is non-cyclic, the canonical quotients G_{n} and the modules M_{n} are not so easily described, even, for example, when G is the alternating group A_{4} or the Klein 4-group. However, the normality of the p-Sylow still strongly affects the modular representation theory of G; we’ll now collect four results we’ll need.

First, observe that we can calculate dim_{𝔽_{p}}(M_{0}) explicitly when G is a pgroup P. Since projective 𝔽_{p}P -modules must be free, ℙ_{𝔽_{p}P} (11 1) ≃ 𝔽_{p}P, and Ω11 1_{𝔽_{p}P} will be the augmentation ideal of 𝔽_{p}P. This is well-known to have the same number of generators r as a module that P has as a group, so ℙ_{𝔽_{p}P} (Ω11 1) will be isomorphic to the direct sum of r copies of 𝔽_{p}P. Hence, dim_{𝔽_{p}}(M_{0}) will equal dim_{𝔽_{p}}(ℙ_{𝔽_{p}P} (Ω11 1)) minus the dimension over 𝔽_{p} of the augmentation ideal, i.e. 1 + (r − 1)|P |. Note the similarity to the Schreier formula; for pro-p groups, one can be derived from the other.

Second, examine the density ρ11 1(𝔽_{p}G). The restriction of the 𝔽_{p}G-module N = 𝔽_{p}G ⊗_{𝔽_{p}A} 11 1_{𝔽_{p}A} to P will be isomorphic to the group ring 𝔽_{p}P, and hence N has no proper non-trivial projective submodule. But the quotient of N by the submodule generated by {(g − 1) ⊗ m | g ∈ P and m ∈ 11 1_{𝔽_{p}A}} is isomorphic to 11 1_{𝔽_{p}G}, so N will be the projective cover of 11 1_{𝔽_{p}G}.

For any finite group Γ, #11 1(𝔽_{p}Γ) = dim_{𝔽_{p}}(ℙ_{𝔽_{p}Γ}(11 1)), cf. [Be98, Lem. 1.7.7 & Prop. 3.1.2] Therefore,

ρ11 1(𝔽_{p}G) = \frac{#11 1(𝔽_{p}G)}{dim_{𝔽_{p}}(𝔽_{p}G)} = \frac{dim_{𝔽_{p}}(N)}{|G|} = \frac{|P |}{|G|} = \frac{1}{|A|}.

Third, Ω^{2}11 1_{𝔽_{p}G}↓_{𝔽_{p}P} ≃ Ω^{2}11 1_{𝔽_{p}P}, so our first observation allows us to explicitly calculate dim_{𝔽_{p}}(Ω^{2}11 1_{𝔽_{p}G}) to be 1 + (r − 1)|P |. This can be seen through Ribes’ result in the second paragraph of this section, since in either case the Frattini module will be isomorphic to ker(ϕ)/Φ(ker(ϕ)). Alternatively, and equivalently, one could use representation theory to prove this. We just saw that ℙ_{𝔽_{p}P} (11 1) ≃ ℙ_{𝔽_{p}G}(11 1)↓_{𝔽_{p}P}; it turns out that this isomorphism preserves the two modules’ radical series, i.e. the product of the module with the successive powers of the Jacobson radical of the group algebra. For example, see [Sem02].

Finally, remember Maschke’s theorem:

Theorem 4.1 (Maschke [Be98, Cor. 3.6.12]). If a finite group A has order relatively prime to the characteristic of the field k, then kA is semisimple.

In other words, 𝔽_{p}A-modules will have trivial cohomology, since there can be no non-split exact sequences of 𝔽_{p}A-modules.

§5. Automorphisms of free groups.

Note that if a pro-p group is (topologically) finite generated, Serre [Ser75] has shown that all of its subgroups having finite index are open; Anderson [A76, Thm. 3] uses, and extends, the proof. Consequently, any automorphism of a (topologically) finitely generated pro-p group will be continuous.

Theorem 5.1. If a non-trivial subgroup A of the group of continuous automorphisms of a non-cyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points Fix_{F} (A) has infinite rank.

Proof. The intuition is that the Density Theorem will force the rank of Fix_{ker_{n}}(A) to be bounded below by a fixed proportion of the rank of ker_{n}, but this growth in its rank is too much for the Schreier formula to allow if Fix_{F} (A) were to have finite rank. To make this concrete, we need some notation.

Let G be the semi-direct product F/Φ(F)>◁A: since Φ(F) is characteristic in F, A will act canonically on F/Φ(F) and we can regard α ∈ A as acting via conjugation — ^{α}x := αxα^{−1} = α(x). The universal p-Frattini cover _{p}G̃ of G is just F >◁A with ker_{0} = Φ(F). We define ker_{n}, G_{n}, and M_{n} as before, and for convenience write F_{n} in place of F/ ker_{n}. Since the p-Sylow F/Φ(F) of G is non-cyclic, the theorem of Griess-Schmid shows that the condition for the Density Theorem holds. By a theorem of Philip Hall [Ha63, Thm. 12.2.2] , any group, having order prime to p, of automorphisms of a pro-p group P must act faithfully on P/Φ(P): Fix_{F_{n}}(A) will be a proper subgroup of F_{n} for all n.

Consider the possibility that F has infinite rank, but Fix_{F} (A) has not. In this case, take any element x of F not fixed by A and consider the closed subgroup of F generated by the union of the orbit of x under A and a finite set of (topological) generators of Fix_{F} (A). As a closed subgroup of a free pro-p group, this will also be free pro-p, but now of finite rank and with a nontrivial quotient of A acting faithfully as continuous automorphisms. But then Fix_{F} (A) would be the subgroup of fixed points of A inside this finite rank free pro-p group. So, assume that F and Fix_{F} (A) have finite ranks r > 1 and s, respectively.

Let us first see that Fix_{ker_{n}}(A)/Fix_{ker_{n+1}}(A) ≃ Fix_{M_{n}}(A); this is equivalent to Fix_{F_{n}}(A) forming a surjective system whose projective limit is Fix_{F} (A). (Note that Fix_{ker_{n}}(A) is just Fix_{F} (A) ∩ ker_{n}.) Given x in Fix_{F_{n}}(A), consider an element x̂ of the preimage of {x} under the natural quotient map φ_{n} : F_{n+1} ↠ F_{n}. The assignment α ↦ a_{α} := ^{α}x̂x̂^{−1} is a 1-cocycle for A with values in M_{n}: ^{α}a_{β}a_{α}a_{αβ}^{−1} = 1, writing M_{n} multiplicatively. Since (|A|, p) = 1, Maschke’s theorem applies and M_{n}↓_{𝔽_{p}A} has trivial cohomology. Hence, there exists a μ ∈ M_{n} such that a_{α} = ^{α}μμ^{−1} for all α in A. The element μ^{−1}x̂ of φ_{n}^{−1}({x}) is then fixed by A.

Since Fix_{F} (A) is a closed subgroup of the free pro-p group F, it must also be free, and so we can use the Schreier formula to compute the rank of its subgroups Fix_{ker_{n+1}}(A):

rank of Fix_{ker_{n+1}}(A) = 1 + (s − 1)|Fix_{F} (A)/Fix_{ker_{n+1}}(A)|
= 1 + (s − 1)|Fix_{F} (A)/Fix_{ker_{n}}(A)||Fix_{M_{n}}(A)|
≤ 1 + (s − 1)|F_{n}||Fix_{M_{n}}(A)|.

As noted before, the density ρ11 1(𝔽_{p}G) is 1/|A|. By the Density Theorem,

lim_{n→∞} \frac{log_{p} |Fix_{M_{n}}(A)|}{log_{p} |M_{n}|^{1/|A|}} = lim_{n→∞} \frac{dim_{𝔽_{p}}(Fix_{M_{n}}(A))}{\frac{1}{|A|} · dim_{𝔽_{p}}(M_{n})} = |A| · ρ11 1(𝔽_{p}G) = 1.

Since |A| > 1, there must then exist a real number ε ∈ (0, 1) and a positive integer N such that |Fix_{M_{n}}(A)| < |M_{n}|^{ε} for all n > N. Now use the identities dim_{𝔽_{p}}(M_{n+1}) = 1 + (r − 1)|F_{n+1}| and |F_{n+1}| = |F_{n}| · |M_{n}|, and the fact that |F_{n}| and |M_{n}| increase monotonically and without bound, to get:

lim_{n→∞} \frac{rank of Fix_{ker_{n+1}}(A)}{dim_{𝔽_{p}}(M_{n+1})} ≤ lim_{n→∞} \frac{1 + (s − 1)|F_{n}||Fix_{M_{n}}(A)|}{1 + (r − 1)|F_{n}||M_{n}|}
= lim_{n→∞} \frac{(s − 1)|Fix_{M_{n}}(A)| · |M_{n}|^{−ε}}{(r − 1)|M_{n}|^{1−ε}}
= 0.

Another use of the Density Theorem gives a contradiction:

lim_{n→∞} \frac{rank of Fix_{ker_{n+1}}(A)}{dim_{𝔽_{p}}(M_{n+1})} ≥ lim_{n→∞} \frac{dim_{𝔽_{p}}(Fix_{M_{n+1}}(A))}{dim_{𝔽_{p}}(M_{n+1})}
= ρ11 1(𝔽_{p}G)
= \frac{1}{|A|}
> 0.  □

We can actually do better in the infinite rank case than this proof might indicate. A slight generalization of [BaFr02, Prop. 5.3] to profinite p-Sylows will show that even if A fixes no non-trivial element of F/Φ(F), a possibility even if F has infinite rank, then the rank of Fix_{Φ(F)/Φ^{2}(F)}(A) will already be infinite.

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[Ser02] Serre, J. P., Galois cohomology, Translated from the French by Patrick Ion and revised by the author. Corrected reprint of the 1997 English edition. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.