The origins of Coclass Theory

In 1980, Leedham-Green and Newman introduced the invariant coclass to the theory of groups of prime-power order and they proposed five far-reaching conjectures related to it. Their work has initiated a deep and fruitful research project in group theory that is still ongoing today. We outline the main results of this celebrated article, we describe the history leading to it, and we survey some highlights of work inspired by these results.


Introduction
In 1980, Leedham-Green and Newman published "Space groups and groups of prime-power order.I", see [45], in the Archiv der Mathematik.In this article, they considered a new grouptheoretic invariant for groups of prime-power order, called coclass, and they formulated five conjectures associated to the groups of a fixed coclass.These conjectures have inspired a wide range of interesting research.Remarkably, not only did all conjectures turn out to be true, their proofs also led to a new structure theory of p-groups and a new approach towards a classification up to isomorphism.It is fair to say that the Coclass Article was a turning-point for the theory of finite p-groups.
We have been invited to review this celebrated article, with the aim to briefly explain its main results, to describe the developments that led to the Coclass Conjectures, and to outline the ongoing impact on p-group theory.Many of the historic details mentioned below are from an interview with Charles Leedham-Green and Mike Newman that we have held in November 2022.
In the remainder of this section, we recall the main results of the Coclass Article.

Coclass
The coclass of a group of prime-power order p n and nilpotency class c is defined as n − c.With the exception of groups of order p, which have coclass 0, the coclass of a finite p-group is an integer between 1 and n − 1.The groups of coclass 1 are exactly the groups of maximal class.These groups have been studied extensively in the literature (we provide more details below), and their intricate structure theory was one of the main incentives for Leedham-Green and

Space Groups
The Coclass Article also contains a first investigation of the soluble infinite pro-p-groups of finite coclass, and it exhibits a link to the theory of space groups.Recall that a space group is an extension of a translation subgroup T ∼ = Z d by a finite point group P that acts faithfully on T .The theory of space groups has a long history in group theory with many interesting applications, for example in crystallography.A p-adic space group is a variation of this: it is an extension of a translation subgroup T ∼ = Z d p , where Z p denotes the p-adic integers, by a point group P of p-power order that acts faithfully on T .Leedham-Green and Newman proved the following in [45,Section 4].
Theorem 1.Let G be an infinite soluble pro-p-group of coclass r.The hypercentre H of G is finite of order p h with h < r.The quotient G/H is an infinite soluble pro-p-group of coclass r − h, and it has the structure of a p-adic space group.

The history of the Coclass Article
What insights have led to the proposal of the invariant "coclass" and the formulation of the Coclass Conjectures?The following history combines some collected facts with comments from the personal interview with Charles Leedham-Green and Mike Newman.
Charles Leedham-Green received his DPhil in 1966 in Oxford and moved to Queen Mary College (London, England) shortly thereafter, where he started a collaboration with his also newly arrived colleague Sue McKay.Huppert's first book on finite groups [33] had just appeared, and they read about Blackburn's and Alperin's results on p-groups of maximal class [2,6].This inspired Leedham-Green and McKay to work on p-groups of maximal class, which resulted in a sequence of papers [38][39][40][41] published between 1976 and 1984.
Mike Newman completed his PhD in 1960 in Manchester.After his PhD, Newman took up a position at the Australian National University (Canberra, Australia).His interests included pgroups and later also computations.At the Australian National University he was involved in the development of the p-group generation algorithm, see [55] for an early outline.Computational experiments with this algorithm were very important for many advances in Coclass Theory (and beyond).

Writing the Coclass Article: 1974-1980
Leedham-Green and Newman met at the ICM 1974 in Vancouver where both of them gave a 'short communication': Newman talked about Groups of exponent four and Leedham-Green reported On p-groups of maximal class.As both shared an interest in p-group theory, they naturally started a collaboration.In his 1990 paper [56, p. 52], Newman recalled that the lecture of Leedham-Green was the first time he has seen a coclass graph for groups of maximal class.At that time, it was known that the graph G(p, 1) has a unique maximal infinite path, and the inverse limit of the groups on this path yields the infinite pro-p-group G p of maximal class.Specifically, G p is isomorphic to a semidirect product P ⋉ T where P is a cyclic group of order p acting via multiplication by a p-th root of unity on the (p − 1)-dimensional p-adic lattice T .
In 1976, Leedham-Green visited Newman in Canberra.On the way out to Canberra, it occurred to Leedham-Green that the infinite pro-p-group associated with G(p, 1) can be gener-alised by replacing P by a cyclic group of order p n , acting via multiplication by a p n -th root of unity on a (p − 1)p n−1 -dimensional p-adic lattice.Upon arrival in Canberra, Leedham-Green discussed this generalisation with Newman who recognised this type of construction as a padic variation of a space group.Newman had just returned home from a visit to the RWTH Aachen, where Neubüser and his collaborators Brown, Bülow, Zassenhaus, and the crystallographer Wondratschek were writing a book on the classification of space groups in dimension 4. Newman transferred many results for integral space groups to p-adic space groups.Similar to how G p yields an infinite path in G(p, 1), the newly constructed infinite groups gave rise to infinite families of finite p-groups, all having the same coclass.This combination of insights gave birth to the new invariant!Conjectures A and B generalise results for p-groups of maximal class and have been influenced by the work of Wiman [69,70], Blackburn [6], Alperin [2], Miech [50][51][52][53], Shepherd [66], and Leedham-Green and McKay [38][39][40][41].Particularly relevant to Conjecture B is a theorem by Alperin that states: For every p, there is a constant m such that every p-group of maximal class has derived length at most m.
Conjectures C, D, and E are of a different nature and rely on the new connection between p-groups and space groups.Leedham-Green wrote in [36]: These conjectures did not come out of the blue.They were first formulated (inaccurately) from the insights obtained from my work with Susan McKay on p-groups of maximal class (that is, of coclass 1), which gave us a proof of the conjectures in this case.That the conjectures appeared in the correct form (and I am still surprised at how accurate they were) was due to M. F. Newman's insight into the structure of space-groups.

Immediate Reactions: 1975-1984
The results and conjectures of the Coclass Article had been talked about prior to the submission in 1979.Many researchers were eager to join the project, and some had already proved some interesting results.It was a time similar to the gold-rush, and it must have been extraordinarily exciting.
The suffix "I" of the Coclass Article suggests that the paper was meant to be the first of a series.Indeed, immediately after its publication, a sequence of papers appeared in print.The first two of them (called "II" and "IV") where written at the RWTH Aachen [31,32].The first by Finken, Neubüser, and Plesken provides a solution to the isomorphism problem of the p-adic space groups with fixed coclass.The second by Felsch, Neubüser, and Plesken determines an infinite family of counterexamples to the class-breadth conjecture via quotients of p-adic space groups.Part "III" of the series is mentioned in [32], but was never published.The other articles of the series (called "V", "VI", etc.) were aimed at proving the Coclass Conjectures; we provide more details below.
Very little was known beyond the detailed structure theory of the p-groups of maximal class and thus more explicit investigations were of interest.In 1975, James [34,35] started the investigation of the groups in G(2, 2).He employed algorithmic methods and even early computers to solve the isomorphism problem for these 2-groups.This was one of the starting points for Ascione [3][4][5], a PhD student of Newman, to consider the 3-groups of coclass 2. Her thesis (137 pages long) provided interesting insights into G(3, 2).

The proof of the Coclass Conjectures
The proofs of the Coclass Conjectures emerged in the years 1980-1994.In many cases, the proofs or at least significant ideas for proofs were around years before they were finally published.Hence, the publication dates often do not accurately reflect the historic events.A full account of the proofs is available in the book by Leedham-Green and McKay [42].

The proof of Conjecture E
At the time when the Coclass Article was written, a proof of Conjecture E was already in the air.It follows from Theorem 1 that Conjecture E is equivalent to the claim that there are only finitely many isomorphism classes of soluble p-adic space groups of fixed coclass.Earlier, around 1911, Bieberbach had proved that there are only finitely many isomorphism classes of integral space groups in any fixed dimension.Thus, it remained to study the relationship between soluble p-adic space groups and integral space groups, and to describe how the coclass of the group is related to the dimension of the space group.
Shortly after the Coclass Article appeared, Leedham-Green, McKay, and Plesken met at the Mathematisches Forschungsinstitut in Oberwolfach to discuss ideas for a proof.One step in the proof was the determination of all the possible point groups; that is, finite p-groups P that can be embedded into GL(d, Z p ) for some suitable dimension d.A crucial ingredient for this step was Vol'vačev's description [68] of the Sylow p-subgroups of GL(d, Q p ). Leedham-Green and Plesken [46] corrected an error for p = 2, and eventually completed the determination of the possible point groups.
The investigation of the extensions of translation subgroups by point groups and the relation between dimension and coclass turned out to be a more subtle business.Charles Leedham-Green remembers: Some while later I found myself in Aachen, working with Wilhelm Plesken on Conjecture E. We formulated a conjecture that would settle the matter, and tested it in dimension 4, using a computer program commissioned by the Berlin bus company.The buses ran on hydrogen that was stored in rare earth crystals, and the hydrogen reduced their symmetry.So a program was commissioned to compute subgroups of space groups.Using it, the machine found a counter-example to our conjecture.
Despite such problems, Leedham-Green, McKay, and Plesken finally obtained a proof of Conjecture E. The proof appeared in two papers [43,44], distinguishing between even and odd primes.For odd primes, there is only one Sylow p-group in GL(d, Q p ), and the extensions constructed from its subgroups behave in a rather generic way.For the even prime p = 2, this is no longer true and the proof required more work.Explicit bounds on the coclass for such p-adic space groups have later been proved by McKay [49].

The proof of Conjecture C
The next step was the proof of Conjecture C; together with the proof of Conjecture E, this also settled Conjecture D.
A first milestone was Donkin's proof [17] of Conjecture C for primes p ≥ 5.This proof relies on the classification of simple p-adic Lie algebras and the theory of p-adic Chevalley groups.Donkin [17] stated that a proof covering all primes has been suggested by Tits; this proof uses the theory of buildings, but it remained unpublished.
The second milestone was the comparatively short and significantly more elementary proof by Shalev and Zelmanov [64] holding for all primes.It relies on a reduction to analytic groups obtained by Leedham-Green [36].

The proof of Conjecture A
Two independent proofs for Conjecture A have been found by Leedham-Green [37] and Shalev [63].Both proofs emerged around 1990/91 and appeared in print in 1994.Mann has written an excellent 'featured review' of both proofs on the AMS MathSciNet.To underline the significance of Conjecture A, Mann also states: According to a reported saying of N. Blackburn, this conjecture, once proved, will be 'the first general theorem of the theory of p-groups'.
Prior to [37,63], special cases of Conjecture A had been established.For example, McKay [48] proved Conjecture A for so-called uncovered CF-groups.Based on this proof, and using the newly developed theory of powerful p-groups, Mann [47] proved Conjecture A for all uncovered p-groups if p > 2.
Shalev's proof [63] of Conjecture A also uses the theory of powerful p-groups and relies on the results of Mann [47].More precisely, it uses that every finite group of coclass r has a powerful subgroup of index bounded by p and r.Assuming Conjecture A is false, these powerful subgroups in a family of assumed counterexamples are used to produce an appropriate Lie ring that, eventually, contradicts a result of Jacobson on Engel Lie algebras.
Leedham-Green's proof [37] appeared later than Shalev's, but (according to Mann) it was the first proof and it was available to Shalev.It relies on Conjecture C and its proof.Inverse limit constructions are a central step in the proof: Assuming Conjecture A is false, an inverse limit construction of an assumed family of counterexamples produces an infinite pro-p-group G of coclass r.This is p-adic analytic by [36], and thus Conjecture C and its proof have implications on the structure of G.The final contradiction is reached by employing results for so-called settled groups and the Lie rings associated with them.

Beyond the Coclass Conjectures
After the Coclass Conjectures have been proved, the main focus shifted on the detailed structure of the coclass graphs G(p, r).The proof of Conjecture D implies that there are only finitely many maximal infinite paths in G(p, r) for any given p and r.For a group G in G(p, r) denote by D(G) the full subtree of G(p, r) consisting of G and its descendants.If G is a group on an infinite path, then D(G) is an infinite tree.It is called a coclass tree if it contains only one infinite path starting at its root G.It is a maximal coclass tree if there is no proper ancestor H of G in G(p, r) such that D(H) is also a coclass tree.The proof of Conjecture D now implies the following; see also the discussion in [42, Section 11.2]: Theorem D'.The graph G(p, r) consists of finitely many maximal coclass trees and finitely many other groups.
Theorem D' implies that the general structure of a coclass graph is determined by the structure of its finitely many maximal coclass trees.By definition, every coclass tree T has a unique infinite path, S 1 → S 2 → . . .say, starting at its root.The inverse limit S of the groups on this path is the infinite pro-p-group of coclass r associated with this maximal coclass tree.For n ∈ N we define the branch B n of T as the difference graph D(S n ) \ D(S n+1 ); that is, B n consists of all descendants of S n in G(p, r) that are not descendants of S n+1 .
For odd p and any given r, Eick [19] introduced an algorithm to determine, up to isomorphism, the infinite pro-p-groups of coclass r; this can be considered as a constructive version of the proof of Theorem D. Further, Eick [20] determined estimates for the number of isomorphism classes of infinite pro-p-groups of coclass r, and showed that this number grows exponentially with p and r.
Regarding the structure of a coclass tree, the following observation holds.
Remark 1.Each maximal coclass tree T consists of its maximal infinite path S 1 → S 2 → . . .and its branches B 1 , B 2 , . ... Each branch B i is a finite tree with root S i .
By Theorem D' and Remark 1, the general structure of a coclass graph G(p, r) is determined by the structure of the branches of the maximal coclass trees in G(p, r).Understanding these branches is the main focus in coclass theory today.

Skeletons
In their papers on p-groups of maximal class, Leedham-Green and McKay introduced a novel construction for certain groups in the branches of the coclass tree in G(p, 1): the so-called constructible groups.This idea was later generalised to all graphs G(p, r) as described in the book [42].The constructible groups contained in a branch B i form a connected subtree.This subtree, truncated at a depth close to the full depth, is called the skeleton of B i .The following theorem of Leedham-Green shows that the skeletons exhibit the broad structure of the branches, see [42,Theorems 11.3.7 and 11.3.9]for a proof.This can be considered as a proof for a stronger version of Conjecture A.
Theorem A'.Given p, r, there are integers n = n(p, r) and m = m(p, r) such that every pgroup P of coclass r of order at least p n has a normal subgroup N of order at most p m so that P/N is a constructible group.
The explicit definition of constructible groups is technical and goes beyond the scope of this survey.The underlying idea is to consider the groups on the infinite path of a coclass tree.These groups are quotients of the associated inverse limit S, and the infinite pro-p-group S is an extension of a normal subgroup T ∼ = Z d p by a finite p-group P .The group T has a unique maximal S-invariant chain T = T 0 > T 1 > . .., and the groups on the original infinite path are isomorphic to the quotients S/T i for large enough i.The constructible groups are now obtained by 'twisting' the multiplication in T /T i such that the result is a group of class 2. The 'twisting' into a group of class 2 is facilitated via the surjective homomorphisms in The choice of i prescribes the order of the resulting constructible group, whereas the choice of j prescribes the branch in which the group lies.A detailed study of the homomorphism space and the isomorphism problem for constructible groups translates to interesting and non-trivial problems in algebraic number theory; this has been investigated in many publications, most recently in [9,13,16].
Theorem A' implies that a classification of skeleton groups yields a classification of all pgroups of coclass r, up to a small "error-term"; it provides a solution for one of the main aims in Coclass Theory, see the quote in Section 1.1.

The graph G(5, 1) and computations
Computational group theory had a major influence in Coclass Theory.For example, finite parts of coclass graphs can be computed with the p-group generation algorithm, which was first developed by Newman [55] and later significantly extended by O'Brien [59].Computer experiments with the p-group generation algorithm were crucial for formulating many of the most recent conjectures on the structure of G(p, r).
The graph G(2, 1) is illustrated in [45] based on the well-known result that the number of 2-groups of maximal class of order 2 n is 3 provided that n ≥ 4. The graph G(3, 1) can be read off from the determination by Blackburn [6] of the 3-groups of maximal class.For larger p, Leedham-Green and McKay [38][39][40][41] described many features of G(p, 1).However, extensive details on the precise structure of G(p, 1) were not known around the year 1990.At that time, computational methods were employed to investigate the smallest open case, the graph G(5, 1).Newman's experiments [56] revealed that G(5, 1) is significantly more complex than G(2, 1) and G(3, 1), yet it has a lot of periodic structure.To describe the latter, we need the notion of pruned branches: for a branch B i and an integer k, the pruned branch B i (k) is the full subtree of B i of all vertices of distance at most k from the root.The following is a simplified variation of [15,Conjecture IV].
Conjecture (G(5, 1)).If i ≥ 9, then the branches B i of the unique maximal coclass tree T in G(5, 1) have depth i + 1, and the following hold: If this conjecture is true, then G(5, 1) can be constructed from a finite subtree using two types of periodicity patterns, both with period 4. The first pattern is part (a): it moves parallel to the infinite path in G(5, 1) and describes how a large pruned part of B i can be embedded into B i+4 .The second pattern is (b): it moves vertically to the infinite path and describes how the "feet" of the branches growth.
Part (a) and a weaker form of part (b) have been proved by Dietrich [11,12], while the complete proof is still open today.

The graphs G(2, r) and Conjecture P
As often in mathematics, the case p = 2 behaves differently to odd primes.The most remarkable difference is that all the skeletons of a maximal coclass tree in G(2, r) are trivial: they only contain a single vertex, the root of the associated branch.Theorem A' now implies that the branches in G(2, r) all have bounded depth, that is, for every coclass r, there is a constant c such that every branch in G(2, r) has depth at most c.This is not true in general for coclass trees with p > 2; for example, see the conjecture on G(5, 1) above.This shows that the graphs G(2, r) have a much simpler structure than most of the graphs G(p, r) for odd primes.
Newman and O'Brien [58] used computational tools to investigate the coclass trees in G(2, r) for r ≤ 3.They classified the groups in G(2, 3) and proposed the following conjecture.
Conjecture P. Let T be a coclass tree in G(2, r) with branches B 1 , B 2 , . ... There exist integers d and n such that B i ∼ = B i+d for each i ≥ n.
This conjecture was proved almost immediately by du Sautoy [18] in the following generalised form.The dimension of a coclass tree is the dimension of the p-adic space group associated to the infinite path in the tree.
Theorem P'.Let p be an arbitrary prime, let r, k be positive integers, and let T be a coclass tree in G(p, r) with branches B 1 , B 2 , . . .and dimension d.There exists an integer n = n(p, r, k) such that B i (k) ∼ = B i+d (k) for all i ≥ n.Du Sautoy's proof for Theorem P' is based on model theory and the theory of zeta-functions; it is non-constructive and it does not yield any bounds for n.A second proof for Theorem P' was given by Eick and Leedham-Green [27]; their proof is constructive and has the additional feature that it yields an explicit analysis of the structure of the groups in B i (k), as well as an explicit bound for n = n(p, r, k).

The graph G(3, 2) and Conjecture W
The 3-groups of coclass 2 were already investigated by Ascione, Havas, and Leedham-Green [3,4] around the time when the Coclass Article was written.Additionally, Leedham-Green wrote detailed notes on G(3, 2) which were intended to be Part "III" in the series of papers started by the Coclass Article.These notes were revisited in 2001 by Eick, Leedham-Green, Newman, and O'Brien.Their paper [28] contains the following result.
(a) The graph G(3, 2) consists of 16 maximal coclass trees.Among the associated infinite pro-3-groups, there are six that are central extensions of the infinite pro-3-group of coclass 1 and there are ten that are 3-adic space groups.(b) Among the 16 maximal coclass trees, 12 have bounded depth; their structure is fully described by Theorem P'.The remaining four maximal coclass trees correspond to 3-adic space groups whose point group is cyclic of order 9 or extraspecial of order 27.
The main part of [28] was concerned with a detailed description of the four maximal coclass trees that were not of bounded depth.These descriptions all rely on the full determination of the skeletons of the branches.Based on this investigation of G(3, 2), the available experimental evidence for G(5, 1), and the periodicity results in [12], the following conjecture was proposed Conjecture W is still open, and proving it is one of the main aims of coclass theory.The main problem is to find a generic description for Q based on P .The group Q is often called a periodic parent for P .

The graphs G(p, 1) and Galois trees
Let T p denote the unique maximal coclass tree in G(p, 1).Then T p has branches of bounded depth for p ∈ {2, 3}, it has branches of growing depth but bounded width for p = 5, and it is growing wildly for p ≥ 7. The growth for p ≥ 7 has recently been investigated by Dietrich and Eick [13] and by Cant, Dietrich, Eick, and Moede [9].
For a group G in the skeleton of a branch in T p we define its Galois order o(G) as the p A subtree O of a skeleton S i of a branch B i in T p is a Galois tree of order o if all groups in O have Galois order o and O is a subtree that is maximal with this property.Dietrich and Eick [13, Theorem 1.1] determined the Galois trees with Galois order p − 1 in G(p, 1) as follows.
Theorem (Galois trees).Let p ≥ 5 and ℓ = (p − 3)/2.Each skeleton in G(p, 1) has ℓ Galois trees with maximal Galois order p − 1.Their roots G 1 , . . ., G ℓ have depth 1 in the skeleton, and their leaves have full depth in the skeleton.A group of non-maximal depth m in the Galois tree with root G i has either 1 or p descendants; the latter case occurs if and only if m is even with (m mod (p − 1)) / ∈ {0, p − 1 − 2i}.
Cant, Dietrich, Eick and Moede [9] generalised this result to Galois trees with arbitrary Galois order.
The main motivation behind investigating Galois trees is indicated in Dietrich's work [12] and in [13]: it is believed that Galois trees might facilitate a method to determine a periodic parent Q for a group P (as proposed in Conjecture W).More precisely, the hope is that periodic parents can be obtained by moving d steps up in a Galois tree of B i+d and then taking the corresponding group in B i .It is known that moving d steps up in a skeleton will, in general, not provide a periodic parent, see Saha [62, Section 6.2.7].

Coclass families
The periodicity result by Eick and Leedham-Green [27] partitions the infinitely many groups in a sequence of pruned branches B 1 (k), B 2 (k), . . .into finitely many coclass families.For this, let n and d be as in Theorem P' and start with a group G in B i (k) with i ∈ {n, n+1, . . ., n+d−1}.Then the associated coclass family is defined as the sequence of groups that can be reached by taking all the images of G under the graph isomorphisms B i (k) → B i+d (k) → B i+2d (k) → . . .as defined in the proof of Theorem P' by Eick and Leedham-Green [27] which also shows that all the groups in one such family can be described by a single parametrised presentation.Wellknown examples of this type of presentation are the coclass families in G(2, 1): these are the families of dihedral groups D 2 n , quaternion groups Q 2 n and semidihedral groups SD 2 n and they can be described by the following parametrised presentations: = x, y | x 2 n−1 = 1, y 2 = x 2 n−2 , x y = x −1 (n ≥ 4), SD 2 n = x, y | x 2 n−1 = 1, y 2 = 1, x y = x 2 n−2 −1 (n ≥ 4).

Other algebraic objects
As final comment, we mention that the invariant "coclass" has also been defined for other algebraic structures, such as nilpotent Lie algebras, nilpotent associative algebras, and semigroups, see for example [10,29,30,61].These investigations have also revealed interesting results, periodicity patterns, and various conjectures.

Open problems and further research
We list a few open problems which we believe are worth to follow up.Many details and background on these will be available in the forthcoming book [14].
Problem 1. Problem 17.63 in the Kourovka notebook.Prove that if p is an odd prime and s a positive integer, then there are only finitely many isomorphism classes of p-adic space groups of finite coclass with point group of coclass s.

[ 28 ,
Conjecture W]: Conjecture W. Let T be a coclass tree in G(p, r) with branches B 1 , B 2 , . . .and dimension d.There are integers n, k such that for all i ≥ n the following hold: (a) B i (k) ∼ = B i+d (k), and (b) for each P of depth k in B i+d there exists Q of depth k − d in B i such that D(P ) ∼ = D(Q).

Problem 2 .Problem 3 .
Problem 14.64  in the Kourovka notebook and Problem 3 in Shalev's survey[65].Complete the description of G(5, 1) and the classification of the 5-groups of coclass 1 via coclass families.More general than Problem 2 is: Complete the description of G(p, 1) for p > 5 and the classification of the p-groups of maximal class via coclass families.Problem 4.Even more general than Problem 3: Prove Conjecture W.