New Lie products for groups and their automorphisms

1 Introduction

Since early work by Magnus, it was known that the lower central series could be used to make a correspondence with free groups and graded Lie rings. This approach became a mainstay of nilpotent groups in part owing to Graham Higman’s promotion in lectures such as “Lie ring methods in the theory of finite nilpotent groups”, which he delivered to the International Congress of Mathematics in 1958 [7]. Already Lazard and Mal’cev had exploited this and even more subtle connections; see [9, Chapter 3] for an excellent modern survey. Higman’s point was to popularize connections to Lie rings “… less precise, but much more generally applicable” [7, p. 307]. The spirit of the present article is similar.

This article points out a few ingredients of group/graded Lie ring correspondences that perhaps could have been known at the time but seem largely overlooked in the literature since. First we observe that the upper central series has an appropriate dual role, leading to Lie module upon which the associated graded Lie ring acts. Second we show that in general one can refine gradings, and the upper and lower series of a group. We call descending generalization a filter and the ascending generalization a layering. The result of such refinement is to further refine inductive studies of groups. As it stands, the common concept of nilpotence degree clusters most finite group theory into a single family of groups of nilpotence class 2. Fragmenting this huge family of groups by whatever means helps when studying these groups in detail, for example in enumeration and isomorphism studies [2, 12, 14, 13, 17]. With such applications in mind, we point out how to lift our filters and layerings of a group 𝐺 to Aut⁡(G) and link the associated graded Lie algebra of Aut⁡(G) to the graded derivation algebra of the Lie algebra for 𝐺. This serves in a way to linearize larger parts of the automorphism group of a group.

For simplicity, we first state our results for the traditional Z+ grading which later we relax to gradings by arbitrary commutative monoids. As usual, the relationship between groups and Lie rings is born of simple ingredients relying on the general observations that group commutators [x,y]=x-1⁢xy=x-1⁢y-1⁢x⁢y obey relations that mimic the distributive, alternating, and the Jacobi identities,

Now consider the usual lower central series where γ1=G and γi+1=[G,γi]. Thus we may define L*⁢(γ)=L*⁢(γ⁢(G))=⊕i>0γi/γi-1 and also the following operations on L*⁢(γ) (for x∈γi, set x¯=x⁢γi+1):

This gives L*⁢(γ) the structure of a graded Lie ring; see [9, Chapter 3]. Perhaps known but certainly underused is the following related product. Consider the usual upper central series where ζ0=1 and ζi+1 is the centralizer of G/ζi. We observe the upper central series can be used to define graded modules of L*⁢(γ) as follows.

For torsion free groups, the use of R=Q=Q is a natural choice. For finite groups, if we use R=Z and Q=Q/Z, then hom⁡(ζi/ζi-1,Q/Z)≅ζi/ζi-1, and for 𝑝-groups, we might prefer Q=Zpe or Zp∞. So we prove the following.

We will make both these results far more general below. Our further connection between central factors and Lie algebras relates to the structure of automorphism groups. In general, we expect no relationship to exist between the central series of a group and its automorphism group (consider G=Zpn and Aut⁡(G)≅GL⁡(d,p)). However, we argue the more appropriate Lie algebra to explore in connection to Aut⁡(G) is the graded derivation ring Der⁡(L⁢(γ)). A hint of this might be recognized from the family of automorphisms 𝜙 on 𝐺 that are the identity on G/ζ1. Such “central” automorphisms induce a linear map G/γ2→ζ1 by γ2⁢x↦x-1⁢(x⁢ϕ). The following theorem can be seen as generalizing central automorphisms but retaining the vital Lie structure. Stated just for the traditional central factors, for any groups 𝐺 and 𝐴, and a homomorphism f:A→Aut⁡(G), define

Both L⁢(γ) and Der⁡(L⁢(γ)) are objects whose structure we can capture through linear algebra and related efficient algorithms as in [3]. Meanwhile, it is difficult to construct a generating set for Aut⁡(G) in most practical settings. So, in this light, Theorem 2 offers a practical approach to learn of some of the structure of Aut⁡(G) without knowing generators for Aut⁡(G).

Theorem 1 and especially its corollary appear classical to us, though we have not discovered them in the literature so far. So we give a synopsis. The mechanics came from a largely unrelated setting. We imitate Whitney’s cap-product from algebraic topology; cf. [6, Section 3.3]. The cap-product is a mixture of homology and cohomology groups. Likewise, commutation products involving the upper central series factors mix the lower and upper central series factors as follows:

(Here and throughout, we use ↣ to distinguish multilinear maps from linear and other functions.) To make a module, we “shuffle” this product. For example, we create

When the Li⁢(ζ) are finite, hom⁡(Li⁢(ζ),Q/Z)≅Li⁢(ζ) leading to Corollary 1.1, but these isomorphisms are not canonical, so it is more natural to retain the coefficients.

2 Generalized ascending and descending central factors

We aim for a generalization of Theorems 1 and 2 that introduces Lie products for groups that are graded by monoids other than the positive integers. The purpose is to find new structure.

By permitting acyclic gradings, we can first profit from all the known Lie methods with standard gradings, and then add recursive tools to refine the rings and modules by increasing the grading parameter.

Throughout, we fix M=⟨M,+,0,≺⟩ a pre-ordered commutative monoid which in particular requires that, for all s∈M, 0≺s, and for all s,t,u,v∈M, if s≺t and u≺v, then s+u≺t+v. In general, we think of M=Nn with either the point-wise or the lexicographic pre-order. In a group 𝐺, commutators are [g,h]=g-1⁢gh=g-1⁢h-1⁢g⁢h, and for X,Y⊆G,

We liberally assume the usual commutator calculus such as in [16, Chapter 5].

Following [17, Section 3], a filterϕ:M→2G on a group 𝐺 is a function on a pre-ordered commutative monoid M=⟨M,+,0,≺⟩ into the subsets 2G of 𝐺 with the following properties. Each ϕs is a subgroup, and for all 𝑠 and 𝑡 in 𝑀,

Thus the burden of explaining the structure of commutation is shared between the operation and the ordering on the monoid. For instance, it is possible to give a total order, i.e. a series of subgroups, but use indices that come from a non-cyclic monoid so that the commutation between the series terms can be far more intricate than by in usual central series.

To every filter ϕ:M→2G, there is an associated boundary filter ∂⁡ϕ:M→2G given by ∂sϕ=⟨ϕs+t:t≠0⟩. Fix a commutative unital associative ring 𝑅, and define the following 𝑅-modules:

Every filter determines the following 𝑀-graded Lie ring [17, Theorem 3.1]:

Here, x¯s denotes a coset in Ls, etc. Set

Actually, each Ls⁢(ϕ;R) is a natural R⁢[ϕ0/∂0⁡ϕ]-module so the portion at the top of a filter can be regarded as storing operators.

In a dual manner, a layeringπ:M→2G on a group 𝐺 is a function where each πs is a subgroup, and for all 𝑠 and 𝑡 in 𝑀,

Form the boundary ∂⁡π:M→2G in the reversed order as ∂s⁡π=⋂t≠0πs+t. So, if M=N, then ∂i⁡π=πi+1. If G=πM=⟨πs:s∈M⟩, then (2.1) says simply that [G,πi+1]≤πi for every 𝑖. This will be recognized as Hall’s condition for a series to be called “central”. Using general monoids, which may not be well-ordered, it is no longer possible to simply take successors to move up; hence ∂s⁡π is used instead. Also, the flexibility to adjust π0 and πM makes it possible to define such series in settings that are non-nilpotent but still related, e.g. in automorphism groups of nilpotent groups.

These definitions determine abelian groups as follows (proved in Section 3):

To state the main result, we need a version with coefficients. For that, we fix an 𝑅-module 𝑄 and set

While filters give natural ring structures, layerings produce natural modules for these rings. A ring is always better understood with a module.

Observe that, in the case where ϕ=γ and π=ζ, we know that, for all i,j>0, [γi,ζi+j]≤ζj. So Theorem 1 is a consequence of Theorem 3.

The next important thrust is to push the effects of a filter into the automorphism group and thus linearize various properties of automorphisms that once required more difficult inspection. Fix groups 𝐺 and 𝐴, and a group homomorphism A→Aut⁡(G). For x∈G and a∈A, we define [x,a]=x-1⁢xa. For a filter ϕ:M→(2G)A (that is a filter of 𝐴-invariant subgroups), define the function Δ⁢ϕ:M→2A given by

Note Theorem 2 follows from Theorem 4.

3 The layers Lt⁢(π;QR) and the action by Lt⁢(ϕ;R)

Since filters were covered in detail along with examples in [17], we can forgo most of that and focus on the new aspects which concern the interplay with layerings. We begin by confirming the claims that Ls⁢(π;R) is well-defined.

Fix a layering π:M→2G. The universe being described here is between the subgroup π0 and πM=⟨πs:s∈M⟩. For every s,u, s≺s+u, which implies that πs≤∂s⁡π=⋂u≠0πs+u. Hence, for all 𝑠,

Thus πs and ∂s⁡π are normal in πM and Ls⁢(π)=∂s⁡π/πs is central in πM/∂s⁡π. In particular, Ls⁢(π) is abelian. Also notice that ∂⁡π:M→2G, and as 𝜋 is order preserving so too is ∂⁡π. Finally,

In particular, ∂⁡π is a layering as well.

Unlike the case of filters, there is no immediate commutation product to impose on L*⁢(π). Nevertheless, we recognize within these ingredients the mechanics similar to the cap-product in algebraic topology; cf. [6, Section 3.3]. We make up a product of a similar shape.

Straining our metaphors, if (2.2) holds, we say that a filter ϕ:M→2Gsifts a layering π:M→2G.

There is of course a similar map Ls+t⁢(π)×Ls⁢(ϕ)↣Lt⁢(π), but this is nothing more than the negative transpose of the map above since [x,y]-1=[y,x]. A more pressing concern is the awkwardness of the indices in the product – it is not graded but nearly. If our monoid has cancellation, we could pass to the associated Grothendieck group and rewrite the grading as Ls×Lt↣Lt-s, but still we would need to clarify at each stage that t-s∈M in order that the product be defined. We promote a remedy that makes the resulting algebra more natural at the cost of introducing coefficients.

Suppose we have an arbitrary bimap (biadditive map) *:U×V↣W and an abelian group 𝑄. We create a second bimap #:U×homZ⁡(W,Q)↣homZ⁡(V,Q) with the following definition:

Such shuffling of products was introduced for nonassociative division rings (semifields) by Knuth and reformulated in a basis free way by Liebler [10, 11]. Though Knuth called these “transposes”, we use the less overloaded term of “shuffle” as suggested by Uriya First [5]. Here, we use them simply to configure the objects above into familiar vocabulary of Lie modules.

Now let Q=QR be an 𝑅-module for a commutative associative unital ring 𝑅. As defined, Ls⁢(π;Q)=homZ⁡(Ls⁢(π),Q). When we apply a Knuth–Liebler shuffle to the products ∘:Ls(ϕ)×Ls+t(π)↣Lt(π), we arrive at a new product

where, for x¯∈Ls⁢(ϕ), r∈R, f:Lt⁢(π)→Q∈Lt⁢(π;Q), (x¯⊗r)⋅f is the following map Ls+t⁢(π)→Q:

This new product is 𝑅-bilinear and further permits us to take the direct sum over all s,t∈M to create a single graded 𝑅-bimap L*⁢(ϕ;R)×L*⁢(π;Q)↣L*⁢(π;Q). It remains to show this makes L*⁢(π;Q) into a Lie module for L*⁢(ϕ;R).

4 Proof of Theorem 3

To complete the proof of Theorem 3, we must now consider Proposition 3.4 in the presence of the appropriate Knuth–Liebler shuffle.

Fix x¯∈Ls⁢(ϕ), y¯∈Lt⁢(ϕ), r,s∈R, and f:Lu⁢(π)→Q∈Lu⁢(π;Q). Recall 𝑅 is commutative and each · is 𝑅-bilinear. So, for all z¯∈Ls+t+u⁢(π), following (3.3), we find the following holds:

Hence, we have

So in L*⁢(π;Q) is a graded Lie module for L*⁢(ϕ;R).∎

5 Proof of Theorem 4

Now we prove our second claim, Theorem 4, which claims that, for an 𝐴-group 𝐺 with an 𝐴-invariant filter ϕ:M→2G, there is a naturally induced filter

as follows:

It further follows that L*⁢(Δ⁢ϕ) maps naturally into Der⁡(L*⁢(ϕ)).

Fix s∈M. Let a,b∈Δs⁢ϕ. So, for all t∈M and all x∈ϕt,

(using also the fact that 𝜙 is 𝐴-invariant). So Δs⁢ϕ is a subgroup of 𝐴. Indeed, if a∈A, then, as ϕt is 𝐴-invariant, [ϕt,a]≤ϕt=ϕt+0. So a∈Δ0⁢ϕ proving that A=Δ0⁢ϕ.

Now suppose s,t∈M. For each u∈M, a∈Δs⁢ϕ, b∈Δt⁢ϕ, x∈ϕu, we find

So [ϕu,[Δs⁢ϕ,Δt⁢ϕ]]≤ϕs+t+u. Hence,

Finally, suppose that s,t∈M and that s≺t. Take a∈Δt⁢ϕ. For each u∈M, s+u≺t+u, and so ϕs+u≥ϕt+u. Hence, [ϕu,a]≤ϕt+u≤ϕs+u, which proves a∈Δs⁢ϕ. So Δs⁢ϕ≥Δt⁢ϕ. This proves Δ⁢ϕ is a filter.

Next we show that, for each s∈M-0, Δs⁢ϕ acts on each Lu⁢(ϕ)=ϕu/∂u⁡ϕ as the identity. For taking a∈Δs⁢ϕ, [ϕu,a]≤ϕu+s≤∂u⁡ϕ. Now define an action of Δs⁢ϕ on L*⁢(ϕ) so that, for each u∈M-0 and a∈Δs⁢ϕ, x¯⁢Da=[x,a]¯ mapping Lu⁢(ϕ)→Ls+u⁢(ϕ). First we observe that

so that [y⁢x,a]=[y,a]x⁢[x,a]≡[x,a](mod∂s+u⁡ϕ). So, for each 𝑢, we have Da:Lu⁢(ϕ)↦Lu+s⁢(ϕ). Next take x,y∈ϕu. As [x,a]∈ϕs+u, it follows that [[x,a],y]∈ϕs+2⁢u≤∂s+u⁡ϕ. Hence,

modulo ∂s+u⁡ϕ. Therefore, Da is an additive endomorphism of L*⁢(ϕ).

Last we show that Da is derivation of L*⁢(ϕ). For u∈M-0 and x,y∈ϕu,

So Da is a derivation on L*⁢(ϕ) shifting the grading by 𝑠.∎

6 Examples

We close with some examples. In particular, we demonstrate how filters and layerings arise naturally within arbitrary groups and are not confined solely to nilpotent contexts.

Consider first the case when G=H×K. Evidently,

Using an acyclic monoid, we get at the structure of 𝐻 and 𝐾 independently. For example, use M=N2 and

It follows that 𝜙 is a filter sifting the layering 𝜋. More generally, given groups ℋ and each H∈H has a filter ϕH:MH→2H sifting a layering πH:M→2H, we obtain a filter ϕ:∏H∈HMH→2∏H sifting a layering π:∏H∈HMH→2∏H, where

We call this a product filter and respectively a product layering.

6.3 Data from small groups

It had been shown in [17, Theorem 4.9] that, using the ring M⁢(∘), there is a positive proportion of all finite groups which admit a proper non-semi-classical filter refining a semi-classical filter. Although log-scale is the only option by today’s estimation techniques (see [1, Chapter 1]), it still could be the case that, in actual proportions, very few groups benefit from this process. So here we report results of an empirical study of this question.

In [13], Maglione created a series of algorithms for the computer algebra system Magma which permit the physical construction of the M⁢(∘) filter refinements described in Section 6.2. This permitted us to conduct an experiment to measure the proportions of groups where refinement is found, adapting that code to also check for refinements by the other four rings. We found at least 9,523,263 of the 11,758,814 groups (81 %) of order at most 1000 have new non-classical commutation products; Figure 1 plots this proportion. Figure 1 shows the proportion of groups of order at most 1000 which have not only a non-classical Lie product, but in fact a non-semi-classical Lie product.

Figure 1

The proportion of groups of order N≤1000 with newly found characteristic non-classical commutation Lie products.

While certainly in some cases the newly recovered characteristic subgroups will have been knowable by other means, we hasten to mention that most of these groups (97 %) are nilpotent of class 2 and have no other standard characteristic subgroups other than the center and commutator. So we are locating structure that is genuinely new. Table 2 shows a breakdown by ring types for groups of prime power order.

Table 1

Lower bound on the proportion of pn order groups having a non-semi-classical refinement.

	𝑛
𝑝	4	5	6	7
2	57 %	75 %	80 %	97 %
3	60 %	70 %	86 %	93 %
5	60 %	68 %	88 %	86 %
7	60 %	67 %	88 %	81 %
541	60 %	51 %	98 %	—

Table 2

Lower bounds on the proportions of groups of order 2n having non-semi-classical refinements by various methods.

𝑁	24	25	26	27	28	29
Total %	57 %	75 %	80 %	97 %	97 %	79 %
% by M⁢(∘)	43 %	51 %	63 %	81 %	68 %	37 %
% by Der⁡(∘)	57 %	75 %	76 %	92 %	95 %	79 %
% by Cen⁡(∘)	57 %	57 %	60 %	65 %	31 %	5.4 %

7 Closing remarks

We have emphasized groups, but the ideas generalize to nonassociative rings and also to loops by first using the work of [15] to obtain an initial associated graded ring. From there, the source for new gradings will likely benefit from the rings described in Section 6.2. There is enormous symmetry within these rings, and often a discovery by one ring can be found by another, but we have examples within the small group data above which show each is needed.

We close with some speculations as to why the proportions seen in Figure 1 are so high. First, if a group 𝐺 has a non-semi-classical refinement then so does any central extension of 𝐺. This explains some of the proportions for 𝑝-groups of class at least 3. For 𝑝-groups of class 2, we are less certain. We can take very large random polycyclic presentations of 𝑝-groups of class 2, and our methods regularly find nothing of interest. On the other hand, in another model of randomness, such as taking quotients or subgroups of upper triangular matrices, we find almost all such examples admit non-semi-classical Lie product. So the likelihood of encountering examples with interesting structure depends greatly on the origin of the groups of interest. Finally, for non-nilpotent groups, what seems a likely explanation (and occurs in random examples) is that they have relatively large Fitting subgroups, and thus the proportions for nilpotent groups have a strong influence. But this is not the whole picture as for example 99 % of the ≈ 1 million groups of order 768=28⋅3 have a non-semi-classical Lie product which exceeds the proportion of nilpotent groups of lower order. Whatever the reasons, the methods above are pliable and can be used to describe other future complex interactions between subgroups. So it is not important that they remain focused on the rings given in Section 6.2.