The classification of p-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph of associated with those p-groups of maximal class whose automorphism group orders are divisible by . We describe the broad structure of by determining its so-called skeleton. We investigate the smallest interesting case in more detail using computational tools, and propose an explicit version of Conjecture W for for arbitrary . Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.
The investigation of the p-groups of maximal class was initiated by Blackburn , and has had a long history since then; we refer to the book of Leedham-Green and McKay  for details and references. A central tool is the coclass graph associated with the p-groups of maximal class: the vertices of are identified with isomorphism type representatives of the considered groups, and there is an edge if and only if , where is the last non-trivial term of the lower central series of H. Investigating the structure of is an approach towards a detailed understanding (and thus towards a possible classification) of the associated groups.
It is well known that consists of an isolated vertex corresponding to the cyclic group of order , and an infinite coclass tree whose root is an elementary abelian group of order . This tree has a single infinite path starting at its root; the groups on this infinite path satisfy and are the lower central series quotients of the (unique) infinite pro-p-group of maximal class. Proofs of these facts and further background information can also be found in .
We need some graph-theoretic notation to describe in more detail. If there is a path of length k from a group G to a group H in , then H is a (k-step) descendant of G, and G is the (k-step) parent of H; a 1-step descendant is an immediate descendant of G. For the branch of is the subtree generated by all descendants of which are not descendants of . Note that every branch is a finite tree with root , and the whole coclass tree of is partitioned into its branches, which are connected via the infinite path, see Figure 1. In conclusion, the structure of is determined by the structure of its branches.
The depth of a vertex in a rooted tree is its distance from the root; the depth of a rooted tree is the maximal depth of a vertex. For the pruned branch is the subtree of generated by all groups of depth at most k in . It was proved independently by du Sautoy  and Eick and Leedham-Green  that for each there exists such that for all . Thus the pruned branches eventually repeat periodically; we call this the first periodicity.
If , then all branches in have depth 1 and therefore can be described by the first periodicity. For larger p, however, the depth of the branches lies between and for , see for example [4, Theorem 1.2]. Hence, the first periodicity is not capable of describing completely. It remains to understand how the branches grow beyond their pruned versions. The work of Leedham-Green and McKay (see [10, 11] and the references there) sheds some light on this. We call a group in capable if it is not a leaf, and we define the skeleton as the subtree of generated by all capable groups at depth at most in . Leedham-Green and McKay  introduced a construction for certain skeleton groups and they showed how the isomorphism problem for these groups is related to number theory over the p-adic rational numbers; we briefly recall and then extend this in Section 4.
We say that has finite width if the number of groups of fixed depth in is bounded by a constant independent of n. It is known that has finite width if and only if . This indicates that is a special case. Indeed, in this case the branches are small enough to be accessible to computer investigations. Moreover, the associated groups are significantly easier to study theoretically than the groups for larger primes. Further references and a more detailed description of explicit results are given in Appendix A.
The situation changes considerably for . Here the graph has infinite width and its branches are too complex for a detailed (computational) investigation. In particular, it is also an open question whether or not the groups of maximal class can be classified by an investigation of . If such a classification is at all possible, then Coclass Conjecture W (see ) proposes an approach: it suggests that there exists an integer k such that for each the branch can be constructed from using two types of periodic patterns. One of these patterns is the first periodicity, the other could be called a second periodicity. We note that the description of the second periodicity in Conjecture W is rather vague and not as explicit as the first periodicity; this may be part of the reason why it is so difficult to investigate. We emphasise that there is only very little evidence for Conjecture W so far and that all the available evidence is in coclass trees of finite width. We exhibit further details on Conjecture W in Appendix A.
1.1 Main results
We consider an arbitrary prime , which is fixed throughout this paper, and define constants
We use the notation , , defined above. The aim of this paper is to study the subgraph of generated by all groups whose automorphism group order is divisible by d. More precisely, we investigate with a view towards understanding the periodicities proposed by Conjecture W; if there is any chance to prove Conjecture W at all, then it seems useful to understand first the possible periodicities in more detail. We denote by and the subgraphs of and contained in ; these are both subtrees of . Our first result is a complete determination of the skeletons ; see Section 6 for its proof.
The skeleton has groups at depth 1 ; we denote these by .
Let H be a descendant of at depth in . Then H has p immediate descendants in if and only if ; otherwise, H has one immediate descendant in .
Theorem 1.1 exhibits that each skeleton consists of subtrees starting at depth 1 and each subtree has a well-described branching pattern depending on the parameter i of the root of the subtree. The next corollary is an immediate consequence.
Let and . The number of groups at depth e in is at least ; in particular, has infinite width.
We have determined for and by computer, cf. Section 1.2. Based on this, we formulate a conjectural description for for arbitrary . To describe this conjecture, we define the twig of a group G in : this is the subtree of with root G containing all descendants of G which are not descendants of any proper descendant of G in . Thus each group of is contained in exactly one twig and the twigs of are connected by the skeleton of . We continue to denote the groups of depth 1 in a skeleton by and we assume that these are sorted via Theorem 1.1 (b). Note that every group G of depth at least 1 in has exactly one of the groups as parent.
There exists such that for all and for each G of depth e at least 1 in with parent the following holds.
If G is not a leaf in , then the isomorphism type of the graph depends on the index i, on , and on only.
If G is a leaf in , then there exists a group with parent at depth in with .
Conjecture 1.3 suggests that there are three types of twigs: the twigs of the leaves in , the twigs of the roots of , and the twigs of the other skeleton groups. Note that, by definition, has depth at most 1 for every group G in which is not a leaf. We choose l large enough such that the first periodicity holds for all groups of depth 1 in ; then the twigs of the roots in behave periodically for all and thus there are at most different twigs of roots in . Conjecture 1.3 suggests that for there are only different twigs in for groups that are neither roots nor leaves. The twigs of the leaves of are trees of depth at most . Conjecture 1.3 suggests that there are finitely many different twigs of skeleton-leaves; more precisely, if the skeleton has leaves, then Conjecture 1.3 proposes that there are at most different twigs arising for the leaves in all with .
1.2 Computations for
We have computed some branches for (and partial branches for ) using the computer algebra system GAP  and the GAP package AnuPQ which is based on . Figure 2 illustrates Conjecture 1.3 with the branches for . The black parts of the graphs and the bold numbers on the left of vertices describe the skeletons: a number k on the left of a vertex indicates that this vertex and all of its descendants appear k times with the same parent. A number w on the right of a vertex says that this vertex has w immediate descendants in addition to the displayed descendants. The grey parts of the graphs are twigs of the leaves in .
1.3 Structure of the paper
In Section 2 we recall some p-adic number theory; these results are important for defining the groups in the skeleton and for solving their isomorphism problem. In Section 4 we recall the construction of the skeleton groups in , and we consider their isomorphism problem and automorphism groups. In Section 5 we then investigate the skeleton groups in in more detail. In particular, we show how they can be constructed up to isomorphism. In Section 6, we prove Theorem 1.1. Appendix A contains a short survey on known periodicity results for coclass graphs.
2 Some number theory
Throughout the paper, and denote the field of p-adic rational numbers and ring of p-adic integers, respectively. The p-th cyclotomic polynomial
is irreducible; we consider a fixed root θ and define , so that K is a field extension of degree over , with -basis . For with the field automorphism is defined by . The Galois group of K is ; it is cyclic and we fix a generator . The equation order is the maximal order of K; it has as -basis, and a unique maximal ideal . We abbreviate , so that for ; this defines a series of ideals through . For and we denote by the direct sum of n copies of (and not the ideal ). For each non-zero there exist unique and a unit such that ; we call the valuation of w. Note that if are non-zero, then . We extend this definition to non-zero n-tuples , so that if and only if .
The generator σ of the Galois group of K can be considered as a -linear map of K. The next lemma determines its eigenvalues; it is proved in [10, Lemma 2.3] for , but the same proof holds for all .
The eigenvalues of the -linear map are and each eigenspace has dimension 1. If with is an eigenvector of σ, then the corresponding eigenvalue is . For every there exists an eigenvector w of σ with .
2.2 The group of units
Let be the unit group of . For define , and let ω be a primitive d-th root of unity in ; we assume throughout that ω is chosen such that where k is defined by the fixed generator of . It is shown in [13, Satz II.5.3] that
The unit group of is , that is, . In the course of the paper we will need various maps based on these unit groups. The following lemma investigates one of them.
The map , , is a group homomorphism with and .
Since σ generates , the fixed points of σ in K are exactly the elements of the subfield of K, hence . Next, we consider the restriction of χ to . As shown in [13, Satz II.5.5 and p. 146], the additive group of is isomorphic to the multiplicative group via
As this exponential map is compatible with the action of σ, we can translate χ to a map , . Since σ is a diagonalisable -linear map on with eigenvalues , the map ψ is diagonalisable with eigenvalues . Now note that if , then , so ; moreover, Lemma 2.1 implies that there exists an eigenvector of ψ in . In conclusion, we have shown that
and hence is the direct product of and restricted to . Finally, note that and . Thus the result follows from the decomposition of as . ∎
3 The infinite pro p-group of maximal class
The following lemma shows how the structure of K relates to the infinite pro-p-group of maximal class, see [11, Proposition 8.3.2]. From now on, we denote by the additive group of the ring , and let P be the cyclic group of order p generated by θ. We let P act on T by multiplication.
The semidirect product is an infinite pro-p-group of coclass 1.
The unique maximal S-invariant series through T is , where each . Moreover, if , then is the i-th term in the lower central series of S. For define , so that is the unique maximal infinite path in . The automorphism groups of S and its quotients are described in the following lemma from [4, Section 4.2]; it implies that the maximal path is contained in .
Writing elements of S as tuples with and , the following hold:
The natural restriction , , satisfies
A preimage of under π is given by
The kernel of π is generated by , where for we define
If , then the natural projection is surjective and .
4 The skeleton groups in
In this section we describe the construction of skeleton groups and their isomorphism problem, based on results of Leedham-Green and McKay , see also [11, Section 8.2]. A key ingredient in that construction is homomorphisms from the exterior square : this is the -module generated by with such that for all and the following holds: , hence , and , and . The group P acts diagonally on , which defines the action on .
In the following let and . Every surjective homomorphism defines an associative multiplication on via
We denote the resulting group by . It is not difficult to show that has class 2 and derived subgroup . Since f is a P-module homomorphism, the multiplication in is compatible with the action of P, and we can define the group
these groups are called constructible in . Each group is an extension of the natural -module by the group on the infinite path of ; in particular, it is a group of depth e in the skeleton . By [4, Theorem 1.3], the groups are exactly the groups in the skeleton .
4.1 The structure of
The structure of has been investigated by Leedham-Green and McKay. We recall some of the results here for completeness, as we need them in later applications. We refer to [11, Sections 8.2 and 8.3] for further details.
First, , where F is a -module of rank generated by with , and Z is a free -module of rank 1. Let denote the Kronecker-delta and recall that is an element of for . For we define via
For we define via
For an -tuple let
For and let
and define the -matrix B over K as
There exists a unique with .
There exists a unique with .
The matrix B describes a base change from to , that is, .
If , then both and lie in . By Lemma 4.1, forms an -basis for . There exists with , which shows that the set generates , but not as an -module. Nonetheless, the latter generating set plays an important role in the solution of the isomorphism problem, see Section 4.2.
The groups at depth e in can be obtained as , where has image ; thus we define
Note that .
For the next lemma, recall the definition of the valuation given in Section 2.
For let be the i-th row of .
We have and .
If with each , then
(a) The proof of [11, Proposition 8.3.8] shows that , hence B is invertible over K. By Lemma 4.1, the i-th row of satisfies ; it remains to analyse the valuation of . It is straightforward to see that each entry of B has valuation , hence
where U is an matrix with entries in only. Since , this implies that . Moreover,
Since , Cramer’s rule for matrix inverses shows that each entry of lies in . Since , each row and column of contains at least one element in . In conclusion, the valuation of the i-th row of is 0, hence the valuation of the i-th row of is .
(b) Let , . Clearly, , and we have to show equality. If , then follows readily; thus we suppose with and . Note that if and only if
Since for all j, we can assume that : simply replace by a multiple . By part (a), each , so implies , that is, there is a uniquely defined such that
Suppose, for a contradiction, that (4.1) is false, that is,
This means that , and so
In other words, if (4.1) is false, then there are , not all 0, such that
We show that this is not possible; then (4.1) must be true, and then so is the claim of the lemma.
Since the rows of are , equation (4.2) is false if and only if the rows of the matrix
are linearly independent over , which is isomorphic to the field with p-elements. This is the case if and only if M is invertible over , if and only if
is invertible over . It follows from the definition of the entries of B that
and is a Vandermonde matrix with parameters . This yields
Note that each , and all are pairwise distinct. This proves that the determinant of is a unit in ; therefore M is invertible over , which proves that (4.2) cannot be true. Thus (4.1) must hold, and the lemma is proved. ∎
4.2 The isomorphism problem and automorphism groups
Recall that the groups in can be constructed as with . By definition, C is an extension of by , and is a fully invariant subgroup of C. Hence the isomorphism problem and the determination of can be approached using the general ideas for group extensions, see for example .
We investigate under which conditions two elements of define isomorphic groups. For this undertaking, the group homomorphisms
with play an important role. We first recall an action on , motivated by .
The element acts on via
This induces an action of on and on the set of cosets for each .
We explain the origin of this action. Every automorphism acts on via ; if f is surjective, then so is . For the following, recall the notation of Lemma 3.2. If β is in the kernel of the map , then for every . Now let with . If , , and , then a short computation shows that . This implies that is mapped to , and as required. ∎
Lemma 4.3 allows us to formulate a solution to the isomorphism problem for skeleton groups and a description for their automorphism groups. For this purpose, let be defined as above and let
be induced by the natural restrictions. It is easy to show that the kernel of is isomorphic to ; its image is described in  using cohomology. Here we describe the image of in a different way that will be more useful in our setting. Recall from Lemma 3.2 that is an extension of by .
Let and ; let .
The groups and are isomorphic if and only if and lie in the same orbit under the action of as defined in Lemma 4.3.
The group is an extension of the p-group by , where Σ is the stabiliser of in .
(a) Every group of depth e in is an extension of by the root of , where carries the obvious -module structure, see [4, Theorem 3.1]. Such extensions can be described by elements of the second cohomology group , and the isomorphism problem of such extensions can be solved by considering the action of the group of compatible pairs , which consists of pairs of compatible automorphisms of and , respectively, see [4, Section 7.1]. By [4, Theorem 7.1], the isomorphism problem for skeleton groups can be solved by considering compatible pairs defined by , acting on certain cohomology classes defined by surjective homomorphisms . This action is discussed in detail in [3, Section 4.2], and it turns out that one has to consider exactly the automorphisms of S which are defined by elements ; the automorphisms with act trivially. Putting all this together, the statement of the theorem follows. We note that in [4, 3] groups of depth e in are described as extensions of by ; however, as -modules, and applying a suitable -module isomorphism translates the results to our set-up, cf. [3, Remark 1].
(b) The claim follows from known results about automorphism groups of extensions, together with [4, Lemma 5.4]. We use the notation introduced in part (a) and let be a cohomology class defining . It is shown in  that the image of is isomorphic to the stabiliser of γ in , and that the kernel of is isomorphic to the p-group . The proof of [3, Lemma 5.4] now shows that the stabiliser of γ in the group of compatible pairs is isomorphic to the stabiliser of in , where acts as the compatible pair . Note that is the stabiliser of in ; since is surjective (and its kernel acts trivially on ), the claim follows. ∎
Leedham-Green and McKay  also considered the isomorphism problem using a different approach: they considered the homomorphism defined by commutation in a skeleton group, and investigated how this homomorphism changes when one modifies certain generators of the group. Their results [10, Propositions 1.1 and 1.2] are in line with ours and might be used to prove one direction of the isomorphism problem.
5 The skeleton groups in
The automorphism group of a skeleton group with is described in Theorem 4.4, and we have
where Σ is the stabiliser of in . Note that and are both p-groups, and also the subquotient induces a p-group in . Recall that ; by Lemma 4.3, the element acts trivially in , and acts by multiplication with . In particular, ω cannot stabilise any non-trivial element in . In summary, if , then is non-trivial and is an extension of a p-group by a subgroup of . Using the notation of Theorem 4.4, we have proved the following result; recall that is cyclic of order and generated by σ.
If and , then
It remains to determine which units arise in this setting, and also how to solve the isomorphism problem for the groups defined by elements in . For this purpose, the following two sets are important; they are defined for ; recall that :
It follows that for every . Note that the elements of are “global” fixed points, while the elements of are fixed points modulo . Every global fixed point induces fixed points modulo , hence for every and we have
We can now state the main result of this section.
Let where and .
There exist and such that .
If and with , then if and only if and for some .
Our proof of Theorem 5.2 proceeds in several steps and is split up into various lemmas; these are exhibited in the following two subsections.
5.1 Proof of Theorem 5.2 (a)
Throughout this section let and .
If , then
if and only if for some .
By Theorem 4.4, if , then
for some and . By Lemma 5.1, there exists an element with . Write for some , and let . Then
as desired. The converse follows directly from Theorem 4.4. ∎
If , then for some such that for some .
In the next lemma we investigate the set in more detail. Recall that ω acts by multiplication by on , and therefore
For each y, the set is a -sublattice of ; moreover
The action of on defined in Lemma 4.3 extends to an action on the -fold direct product ; in particular, acts via
By Lemma 2.1, is diagonalisable with eigenvalues , and each eigenspace has dimension 1. Thus, the action of on is diagonalisable with eigenvalues and each eigenspace has dimension . We denote by the eigenspace with eigenvalue in .
It follows from (5.1) that , hence and each is a sublattice of . It remains to show the following inclusion: . For this purpose, consider and write with for . Let k be the number of non-zero summands in w. We prove by induction on k that each ; then the assertion of the lemma follows.
If or , then our claim is obviously true. Now suppose that and choose j with . It follows from Lemma 4.3 that both and lie in , hence also their difference lies in . Note that , where each . By construction, u has at most non-zero summands and, by the induction hypothesis, we have for each i. Recall that ; this implies that and hence for all . Since , it follows that for all . In particular, we have , and so as well. Now clearly , which completes the proof. ∎
The next lemma considers and with , as in Lemma 5.4. Recall that every can be written as
for some and .
Let and with , and write for some and . There exists with .
By definition, ; by Lemma 5.5, we can decompose with each . As and , it follows that
As , Lemma 5.5 yields that for each i. Using , it follows that ; in particular, and for all . We can now choose ; we have shown that , thus
As , it follows that . Hence as claimed. ∎
We have if and only if has an eigenvector in with eigenvalue .
Let be an eigenvector of with eigenvalue . By Lemma 2.1, there exists an eigenvector of σ with eigenvalue . Then and
so . Conversely, let . Then is an eigenvector of with eigenvalue . If is an eigenvector of σ with eigenvalue , then and
The next lemma is the last result we need for our proof of Theorem 5.2 (a).
The eigenvalues of on are .
The main tool in the following proof is Lemma 4.2: with Lemma 2.1 it implies that the eigenvalue of an eigenvector of is , where is the valuation defined in Section 2. The eigenvalues on coincide with the eigenvalues of on the elementary abelian quotient of rank ; in particular, this number of different eigenvalues is at most . We show that it is exactly . To prove this, we use the notation of Lemma 4.2 and define an isomorphism
recall that each has valuation . In the following we let act on via ψ; in particular, the action of on is diagonalisable. Our claim is that for each there is, up to scalar multiples, a unique eigenvector of with , and that this eigenvector has eigenvalue . We use induction to prove this claim. We note that the assertion of the lemma follows from this claim.
Base case. For the base case consider . Since the action of on is diagonalisable, there must be an eigenvector with . This eigenvector comes from an eigenvector for some . Recall that . Now
and imply that , see Lemma 4.2, whence the eigenvalue of (and a) is . For the uniqueness, consider an eigenvector with ; as just shown, the eigenvalue is . Without loss of generality, we can assume that . Suppose, for a contradiction, that a and are linearly independent, so that
is an eigenvector of with eigenvalue . This eigenvector comes from an eigenvector for some . But now , so the eigenvalue of b cannot be , a contradiction. (Indeed, if , then forces , so and , which is not possible.) This proves that there is, up to scalar multiples, a unique eigenvector of in of the form with , and that the corresponding eigenvalue is .
Induction hypothesis. Our induction hypothesis now is that for each index there is, up to scalar multiples, a unique eigenvector of in of the form
with , and that the corresponding eigenvalue is . Note that comes from an eigenvector
for some , with
Existence of eigenvector. It follows from the induction hypothesis that there is an eigenvector
of with : if not, then there would be a basis of consisting of and i additional eigenvectors each having 0 as k-th entry for all ; this is not possible as such a set of i vectors cannot be linearly independent. Thus, an eigenvector a as above exists.
Eigenvalue. Our first claim is that the corresponding eigenvalue is . Note that a comes from an eigenvector
for some ; in the following, write . If
then by Lemma 4.2, and it follows that the eigenvalue is . It remains to consider the case , so that the eigenvalue of a is . We show that this is not possible; we achieve this by modifying by our known eigenvectors until we obtain a contradiction. For this purpose, write
and let be minimal with ; such an s exists by Lemma 4.2. Note that since , hence . Since by assumption, this implies . By the induction hypothesis, we know the existence of the eigenvector
of with eigenvalue . Let be an eigenvector of σ with eigenvalue . Now is an eigenvector of with eigenvalue , and that both and correspond to
since . In particular,
Since both , we can replace k by a suitable scalar multiple of k such that , and so ; note that for all j. In conclusion, we have found with such that if and is minimal with , then . (Note that has eigenvalue and , so we must indeed have .) Now we iterate this argument until we find an eigenvector
of with eigenvalue and such that if , then for all j. But then Lemma 4.2 implies that , a contradiction to and . In summary, this proves , hence the eigenvalue of an eigenvector with must be .
Uniqueness. Consider a second eigenvector of with ; as proved in the previous paragraph, the eigenvalue of is . We claim that and a are linearly dependent. Note that a comes from an eigenvector for some with . Similarly, comes from an eigenvector for some with . Suppose, for a contradiction, that a and are linearly independent. Replacing by a suitable scalar multiple, we can assume that , so that , with and , is also an eigenvector of with eigenvalue . This eigenvector comes from , where satisfies ; in fact, we must have since otherwise the eigenvalue of b cannot be . Note that for all we already found eigenvectors with eigenvalue , thus we can use the same construction as in the previous paragraph to obtain from and an eigenvector with eigenvalue , where satisfies : this is not possible since has eigenvalue , that is, , but we have deduced that . This contradiction proves that a and must be linearly dependent. In conclusion, we have proved that, up to scalar multiples, there is a unique eigenvector of of the form with , and that the corresponding eigenvalue is . This completes the induction step. ∎
We have if and only if for some .
We can now prove Theorem 5.2 (a).
5.2 Proof of Theorem 5.2 (b)
Again, we assume that and . Recall the map , , which is discussed in Lemma 2.2.
Let and with . If , then ; if with as in Lemma 5.3, then and are fixed points of modulo .
It is shown in Lemma 5.3 that for some . This implies that
By Theorem 4.4, the element yields an element of the group , and the discussion of the automorphism groups of skeleton groups (in the beginning of Section 5) forces that . Since we have , this yields . In turn, this implies that (and, by duality also ) are fixed points of modulo . ∎
Let with . If , then for some .
Recall that . Thus, by Lemma 5.3, there exists with . By Lemma 5.10, both and are fixed points under . It follows from Lemma 2.2 that the restriction induces an automorphism of ; write with and . We want to show that since this implies that
which proves the lemma.
We start with a more general observation: Let and suppose maps the fixed point of to another fixed point of modulo . Then it follows from
that such a has an image which acts trivially on modulo , and by duality also on modulo .
Since maps the fixed point to the fixed point , it follows that stabilises modulo . Note that , hence also stabilises modulo . Now we iterate this argument: since maps the fixed point of to the fixed point of , modulo , it follows from the general observation that stabilises modulo . By induction, stabilises modulo for every .
There is such that acts trivially on : for example, choose j large enough such that for some x with . It follows from the definition that χ stabilises . Since , the proof of Lemma 2.2 shows that ; this implies that χ induces an automorphism of , which we denote by ψ. Since J is a finite group, it follows that ψ has finite order, say t.
Recall that w as above lies in . If w lies in , then acts trivially on modulo , and there is nothing to show. If , then its coset in J is non-trivial, and follows. But this means that for some . As shown above, stabilises modulo . Since acts trivially on , it follows that w stabilises modulo . ∎
We can now prove Theorem 5.2 (b).