In Notes 5, we saw that the Gowers uniformity norms on vector spaces in high characteristic were controlled by classical polynomial phases .
Now we study the analogous situation on cyclic groups . Here, there is an unexpected surprise: the polynomial phases (classical or otherwise) are no longer sufficient to control the Gowers norms once exceeds . To resolve this problem, one must enlarge the space of polynomials to a larger class. It turns out that there are at least three closely related options for this class: the local polynomials, the bracket polynomials, and the nilsequences. Each of the three classes has its own strengths and weaknesses, but in my opinion the nilsequences seem to be the most natural class, due to the rich algebraic and dynamical structure coming from the nilpotent Lie group undergirding such sequences. For reasons of space we shall focus primarily on the nilsequence viewpoint here.
Traditionally, nilsequences have been defined in terms of linear orbits on nilmanifolds ; however, in recent years it has been realised that it is convenient for technical reasons (particularly for the quantitative “single-scale” theory) to generalise this setup to that of polynomial orbits , and this is the perspective we will take here.
A polynomial phase on a finite abelian group is formed by starting with a polynomial to the unit circle, and then composing it with the exponential function . To create a nilsequence , we generalise this construction by starting with a polynomial into a nilmanifold , and then composing this with a Lipschitz function . (The Lipschitz regularity class is convenient for minor technical reasons, but one could also use other regularity classes here if desired.) These classes of sequences certainly include the polynomial phases, but are somewhat more general; for instance, they almost include bracket polynomial phases such as . (The “almost” here is because the relevant functions involved are only piecewise Lipschitz rather than Lipschitz, but this is primarily a technical issue and one should view bracket polynomial phases as “morally” being nilsequences.)
In these notes we set out the basic theory for these nilsequences, including their equidistribution theory (which generalises the equidistribution theory of polynomial flows on tori from Notes 1) and show that they are indeed obstructions to the Gowers norm being small. This leads to the inverse conjecture for the Gowers norms that shows that the Gowers norms on cyclic groups are indeed controlled by these sequences.
— 1. General theory of polynomial maps —
In previous notes, we defined the notion of a (non-classical) polynomial map of degree at most between two additive groups , to be a map obeying the identity
for all , where is the additive discrete derivative operator.
There is another way to view this concept. For any , define the Host-Kra group of of dimension and degree to be the subgroup of consisting of all tuples obeying the constraints
for all faces of the unit cube of dimension at least , where . (These constraints are of course trivial if .) A -dimensional face of the unit cube is of course formed by freezing of the coordinates to a fixed value in , and letting the remaining coordinates vary freely in .
Thus for instance is (essentially) space of parallelograms in , while is the diagonal group , and is all of .
It turns out (somewhat remarkably) that these notions can be satisfactorily generalised to non-abelian setting, this was first observed by Leibman (in these papers, and also later by personal communication, in which the role of the Host-Kra group was emphasised). The (now multiplicative) groups need to be equipped with an additional structure, namely that of a filtration.
Definition 1 (Filtration) A filtration on a multiplicative group is a family of subgroups of obeying the nesting property
and the filtration property
for all , where is the group generated by , where is the commutator of and . We will refer to the pair as a filtered group. We say that an element of has degree if it belongs to , thus for instance a degree and degree element will commute modulo errors.
In practice we usually have . As such, we see that for all , and so all the are normal subgroups of .
Exercise 2 Define the lower central series
of a group by setting and for . Show that the lower central series is a filtration of . Furthermore, show that the lower central series is the minimal filtration that starts at , in the sense that if is any other filtration with , then for all .
Example 1 If is an abelian group, and , we define the degree filtration on by setting if and for .
Definition 2 (Host-Kra groups) Let be a filtered group, and let be an integer. The Host-Kra group is the subgroup of generated by the elements with an arbitrary face in and an element of , where is the element of whose coordinate at is equal to when and equal to otherwise.
From construction we see that the Host-Kra group is symmetric with respect to the symmetry group of the unit cube . We will use these symmetries implicitly in the sequel without further comment.
Example 3 Let us parameterise an element of as . Then is generated by elements of the form for , and , and for . (This does not cover all the possible faces of , but it is easy to see that the remaining faces are redundant.) In other words, consists of all group elements of the form , where , , and . This example is generalised in the exercise below.
Exercise 3 Define a lower face to be a face of a discrete cube in which all the frozen coefficients are equal to . Let us order the lower faces as in such a way that whenever is a subface of . Let be a filtered group. Show that every element of has a unique representation of the form , where and the product is taken from left to right (say).
Exercise 4 If is an abelian group, show that the group defined in Definition 2 agrees with the group defined at the beginning of this section for additive groups (after transcribing the former to multiplicative notation).
Exercise 5 Let be a filtered group. Let be an -dimensional face of . Identifying with in an obvious manner, we then obtain a restriction homomorphism from with . Show that the restriction of any element of to then lies in .
Exercise 6 Let be a filtered group, let and be integers, and let and be elements of . Let be the element of defined by setting for to equal for , and equal to otherwise. Show that if and only if and , where is defined in Example 2. (Hint: use Exercises 3, 5.)
Exercise 7 Let be a filtered group, let , and let be an element of . We define the derivative in the first variable to be the tuple . Show that if and only if the restriction of to lies in and lies in , where is defined in Example 2.
Remark 1 The the Host-Kra groups of a filtered group in fact form a cubic complex, a concept used in topology; but we will not pursue this connection here.
In analogy with Exercise 1, we can now define the general notion of a polynomial map:
Definition 3 A map between two filtered groups is said to be polynomial if it maps to for each . The space of all such maps is denoted .
Since are groups, we immediately obtain
Theorem 4 (Lazard-Leibman theorem) forms a group under pointwise multiplication.
(From our choice of definitions, this theorem is a triviality, but the theorem is less trivial when using an alternate but non-trivially equivalent definition of a polynomial, which we will give shortly.) In a similar spirit, we have
Theorem 5 (Filtered groups and polynomial maps form a category) If and are polynomial maps between filtered groups , then is also a polynomial map.
We can also give some basic examples of polynomial maps. Any constant map from to taking values in is polynomial, as is any map which is a filtered homomorphism in the sense that it is a homomorphism from to for any .
Now we turn to an alternate definition of a polynomial map. For any and any map Define the multiplicative derivative by the formula .
Theorem 6 (Alternate description of polynomials) Let be a map between two filtered groups . Then is polynomial if and only if, for any , , and for , one has .
In particular, from Exercise 1, we see that a non-classical polynomial of degree from one additive group to another is the same thing as a polynomial map from to . More generally, a map from to a filtered group is polynomial if and only if
for all and .
Proof: We first prove the “only if” direction. It is clear (by using -dimensional cubes) that a polynomial map must map to . To obtain the remaining cases, it suffices by induction on to show that if is polynomial from to , and for some , then is polynomial from to . But this is easily seen from Exercise 7.
Now we establish the “if” direction. We need to show that maps to for each . We establish this by induction on . The case is trivial, so suppose that and that the claim has already been estabilshed for all smaller values of .
By telescoping series, it suffices to establish this when for some face of some dimension in and some , as these elements generate . But then vanishes outside of and is equal to on , so by Exercise 6 it will suffice to show that , where is restricted to (which one then identifies with ). But by the induction hypothesis, maps to , and the claim then follows from Exercise 5.
Exercise 8 Let be integers. If is a filtered group, define to be the subgroup of generated by the elements , where ranges over all faces of and , where are the coordinates of that are frozen. This generalises the Host-Kra groups , which correspond to the case . Show that if is a polynomial map from to , then maps to .
Exercise 9 Suppose that is a non-classical polynomial of degree from one additive group to another. Show that is a polynomial map from to for every . Conclude in particular that the composition of a non-classical polynomial of degree and a non-classical polynomial of degree is a non-classical polynomial of degree .
Exercise 10 Let , be non-classical polynomials of degrees , respectively between additive groups , and let be a bihomomorphism to another additive group (i.e. is a homomorphism in each variable separately). Show that is a non-classical polynomial of degree .
— 2. Nilsequences —
We now specialise the above theory of polynomial maps to the case when is just the integers (viewed additively) and is a nilpotent group. Recall that a group is nilpotent of step at most if the group in the lower central series vanishes; thus for instance a group is nilpotent of step at most if and only if it is abelian. Analogously, let us call a filtered group nilpotent of degree at most if vanishes. Note that if and is nilpotent of degree at most , then is nilpotent of step at most . On the other hand, the degree of a filtered group can exceed the step; for instance, given an additive group and an integer , has degree but step . The step is the traditional measure of nilpotency for groups, but the degree seems to be a more suitable measure in the filtered group category.
We refer to sequences which are polynomial maps from to as polynomial sequences or Hall-Petresco sequences adapted to . The space of all such sequences is denoted ; by the machinery of the previous section, this is a multiplicative group. These sequences can be described explicitly:
Exercise 11 Let be an integer, and let be a filtered group which is nilpotent of degree . Show that a sequence is a Hall-Petresco sequence if and only if one has
for all and some for , where . Furthermore, show that the are unique. We refer to the as the Taylor coefficients of at the origin.
Exercise 12 In a degree nilpotent group , establish the formula
for all and . This is the first non-trivial case of the Hall-Petresco formula, a discrete analogue of the Baker-Campbell-Hausdorff formula that expresses the polynomial sequence explicitly in the form (1).
Define a nilpotent filtered Lie group of degree to be a nilpotent filtered group of degree , in which and all of the are connected, simply connected finite-dimensional Lie groups. A model example here is the Heisenberg group, which is the degree nilpotent filtered Lie group
(i.e. the group of upper-triangular unipotent matrices with arbitrary real entries in the upper triangular positions) with
and trivial for (so in this case, is also the lower central series).
Exercise 13 Show that a sequence
from to the Heisenberg group is a polynomial sequence if and only if are linear polynomials and is a quadratic polynomial.
It is a standard fact in the theory of Lie groups that a connected, simply connected nilpotent Lie group is topologically equivalent to its Lie algebra , with the homeomorphism given by the exponential map (or its inverse, the logarithm function . Indeed, the Baker-Campbell-Hausdorff formula lets one use the nilpotent Lie algebra to build a connected, simply connected Lie group with that Lie algebra, which is then necessarily isomorphic to . One can thus classify filtered nilpotent Lie groups in terms of filtered nilpotent Lie algebras, i.e. a nilpotent Lie algebras together with a nested family of sub-Lie algebras
with the inclusions (in which the bracket is now the Lie bracket rather than the commutator). One can describe such filtered nilpotent Lie algebras even more precisely using Mal’cev bases; see these papers of Mal’cev and of Leibman. For instance, in the case of the Heisenberg group, one has
From the filtration property, we see that for , each is a normal closed subgroup of , and for , the quotient group is connected, simply connected abelian Lie group (with Lie algebra ), and is thus isomorphic to a vector space (with the additive group law). Related to this, one can view as a group extension of the quotient group (with the degree filtration ) by the central vector space . Thus one can view degree filtered nilpotent groups as an -fold iterated tower of central extensions by finite-dimensional vector spaces starting from a point; for instance, the Heisenberg group is an extension of by .
We thus see that nilpotent filtered Lie groups are generalisations of vector spaces (which correspond to the degree case). We now turn to filtered nilmanifolds, which are generalisations of tori. A degree filtered nilmanifold is a filtered degree nilpotent Lie group , together with a discrete subgroup of , such that all the subgroups in the filtration are rational relative to , which means that the subgroup is a cocompact subgroup of (i.e. the quotient space is cocompact, or equivalently one can write for some compact subset of . Note that the subgroups give the structure of a degree filtered nilpotent group
Exercise 14 Let and , and let . Show that the subgroup of is rational relative to if and only if is a rational number; this may help explain the terminology “rational”.
By hypothesis, the quotient space is a smooth compact manifold. The space is a compact connected abelian Lie group, and is thus a torus; the degree filtered nilmanifold can then be viewed as a principal torus bundle over the degree filtered nilmanifold with as the structure group; thus one can view degree filtered nilmanifolds as an -fold iterated tower of torus extensions starting from a point. For instance, the Heisenberg nilmanifold
is an extension of the two-dimensional torus by the circle .
Every torus of some dimension can be viewed as a unit cube with opposite faces glued together; up to measure zero sets, the cube then serves as a fundamental domain for the nilmanifold. Nilmanifolds can be viewed the same way, but the gluing can be somewhat “twisted”:
for all , show that for almost all , that has exactly one representation of the form with , which is given by the identity
where is the greatest integer part of , and is the fractional part function. Conclude that is topologically equivalent to the unit cube quotiented by the identifications
between opposite faces.
Note that by using the projection , we can view the Heisenberg nilmanifold as a twisted circle bundle over , with the fibers being isomorphic to the unit circle . Show that is not homeomorphic to . (Hint: show that there are some non-trivial homotopies between loops that force the fundamental group of to be smaller than .)
The logarithm of the discrete cocompact subgroup can be shown to be a lattice of the Lie algebra . After a change of basis, one can thus view the latter algebra as a standard vector space and the lattice as . Denoting the standard generators of the lattice (and the standard basis of ) as , we then see that the Lie bracket of two such generators must be an integer combination of more generators:
The structure constants describe completely the Lie group structure of and . The rational subgroups can also be described by picking some generators for , which are integer combinations of the . We say that the filtered nilmanifold has complexity at most if the dimension and degree is at most , and the structure constants and coefficients of the generators also have magnitude at most . This is an admittedly artificial definition, but for quantitative applications it is necessary to have some means to quantify the complexity of a nilmanifold.
A polynomial orbit in a filtered nilmanifold is a map of the form , where is a polynomial sequence. For instance, any linear orbit , where and , is a polynomial orbit. The space of
(using the notation from Exercise 15) is a polynomial sequence in the Heisenberg nilmaniofold.
With the above example, we see the emergence of bracket polynomials when representing polynomial orbits in a fundamental domain. Indeed, one can view the entire machinery of orbits in nilmanifolds as a means of efficiently capturing such polynomials in an algebraically tractable framework (namely, that of polynomial sequences in nilpotent groups). The piecewise continuous nature of the bracket polynomials is then ultimately tied to the twisted gluing needed to identify the fundamental domain with the nilmanifold.
Finally, we can define the notion of a (basic Lipschitz) nilsequence of degree . This is a sequence of the form , where is a polynomial orbit in a filtered nilmanifold of degree , and is a Lipschitz function. (One needs a metric on to define the Lipschitz constant, but this can be done for instance by using a basis of to identify with a fundamental domain , and using this to construct some (artificial) metric on . The details of such a construction will not be important here.) We say that the nilsequence has complexity at most if the filtered nilmanifold has complexity at most , and the (inhomogeneous Lipschitz norm) of is also at most .
A basic example of a degree nilsequence is a polynomial phase , where is a polynomial of degree . A bit more generally, is a degree sequence, whenever is a Lipschitz function. In view of Exercises 15, 16, we also see that
or more generally
or more generally
are also degree nilsequences, where is a Lipschitz function that vanishes near and . The factor is not needed (as there is no twisting in the coordinate in Exercise 15), but the factor is (unfortunately) necessary, as otherwise one encounters the discontinuity inherent in the term (and one would merely have a piecewise Lipschitz nilsequence rather than a genuinely Lipschitz nilsequence). Because of this discontinuity, bracket polynomial phases cannot quite be viewed as Lipschitz nilsequences, but from a heuristic viewpoint it is often helpful to pretend as if bracket polynomial phases are model instances of nilsequences.
The only degree nilsequences are the constants. The degree nilsequences are essentially the quasiperiodic functions:
Exercise 17 Show that a degree nilsequence of complexity is Fourier-measurable with growth function depending only on , where Fourier measurability was defined in Notes 2.
Exercise 18 Show that the class of nilsequences of degree does not change if we drop the condition , or if we add the additional condition .
Remark 2 The space of nilsequences is also unchanged if one insists that the polynomial orbit be linear, and that the filtration be the lower central series filtration; and this is in fact the original definition of a nilsequence. The proof of this equivalence is a little tricky, though, and will appear in a forthcoming paper of Green, Ziegler, and myself.
— 3. Connection with the Gowers norms —
We define the Gowers norm of a function by the formula
where is any integer greater than , is embedded inside , and is extended by zero outside of . It is easy to see that this definition is independent of the choice of . Note also that the normalisation factor is comparable to when is fixed and is comparable to .
One of the main reasons why nilsequences are relevant to the theory of the Gowers norms is that they are an obstruction to that norm being small. More precisely, we have
We now prove this theorem, following an argument of Green, Ziegler, and myself. It is convenient to introduce a few more notions. Define a vertical character of a degree filtered nilmanifold to be a continuous homomorphism that annihilates , or equivalently an element of the Pontryagin dual of the torus . A function is said to have vertical frequency if obeys the equation
for all and . A degree nilsequence is said to have a vertical frequency if it can be represented in the form for some Lipschitz with a vertical frequency.
For instance, a polynomial phase , where is a polynomial of degree , is a degree nilsequence with a vertical frequency. Any nilsequence of degree is trivially a nilsequence of degree with a vertical frequency of . Finally, observe that the space of degree nilsequences with a vertical frequency is closed under multiplication and complex conjugation.
Exercise 19 Show that a degree nilsequence with a vertical frequency necessarily takes the form for some and (and conversely, all such sequences are degree nilsequences with a vertical frequency). Thus, up to constants, degree nilsequences with a vertical frequency are the same as Fourier characters.
A basic fact (generalising the invertibility of the Fourier transform in the degree case) is that the nilsequences with vertical frequency generate all the other nilsequences:
Exercise 20 Show that any degree nilsequence can be approximated to arbitrary accuracy in the uniform norm by a linear combination of nilsequences with a vertical frequency. (Hint: use the Stone-Weierstrass theorem.)
More quantitatively, show that a degree nilsequence of complexity can be approximated uniformly to error by a sum of nilsequences, each with a representation with a vertical frequency that is of complexity . (Hint: this can be deduced from the qualitative result by a compactness argument using the Arzelá-Ascoli theorem.)
A derivative of a polynomial phase is a polynomial phase of one lower degree. There is an analogous fact for nilsequences with a vertical frequency:
Lemma 8 (Differentiating nilsequences with a vertical frequency) Let , and let be a degree nilsequence with a vertical frequency. Then for any , is a degree nilsequence. Furthermore, if has complexity (with a vertical frequency representation), then has complexity .
Proof: We just prove the first claim, as the second claim follows by refining the argument.
We write for some polynomial sequence and some Lipschitz function with a vertical frequency. We then express
where is the function
and is the sequence
Now we give a filtration on by setting
for , where is the subgroup of generated by and the diagonal group . One easily verifies that this is a filtration on . The sequences and are both polynomial with respect to this filtration, and hence by the Lazard-Leibman theorem, is polynomial also.
Next, we use the hypothesis that has a vertical frequency to conclude that is invariant with respect to the action of the diagonal group . If we then define to be the Lie group with filtration , then is a degree filtered nilpotent Lie group; setting , we conclude that is a degree nilmanifold and
where are the projections of from to . The claim follows.
We now prove Theorem 7 by induction on . The claim is trivial for , so we assume that and that the claim has already been proven for smaller values of .
(extending by zero outside of , and extending arbitrarily) to conclude that
for values of . By induction hypothesis and Lemma 8, we conclude that
for values of . Using the identity
we close the induction and obtain the claim.
In the other direction, we have
This conjecture has recently been proven by Green, Ziegler, and myself; an announcement of this result, which will contain extensive heuristic discussion of how this conjecture is proven, will appear very shortly, and the paper itself soon after that. For a discussion of the history of the conjecture, including the cases , see our previous paper.
Exercise 21 ( inverse theorem)
- (Straightening an approximately linear function) Let . Let be a function such that for all but of all with . If is sufficiently small, show that there exists an affine linear function with such that for all but values of , where as . (Hint: One can take to be small. First find a way to lift in a nice manner from to .)
- Let be such that and . Show that there exists a polynomial of degree such that , where as (holding fixed). Hint: Adapt the argument of the analogous finite field statement. One cannot exploit the discrete nature of polynomials any more; and so one must use the preceding part of the exercise as a substitute.
The inverse conjecture for the Gowers norms, when combined with the equidistribution theory for nilsequences that we will turn to next, has a number of consequences, analogous to the consequences for the finite field analogues of these facts; see this paper of Green and myself for further discussion.
— 4. Equidistribution of nilsequences —
In the subject of higher order Fourier analysis, and in particular in the proof of the inverse conjecture for the Gowers norms, as well as in several of the applications of this conjecture, it will be of importance to be able to compute statistics of nilsequences , such as their averages for a large integer ; this generalises the computation of exponential sums such as that occurred in Notes 1. This is closely related to the equidistribution of polynomial orbits in nilmanifolds. Note that as is a compact quotient of a locally compact group , it comes endowed with a unique left-invariant Haar measure (which is isomorphic to the Lebesgue measure on a fundamental domain of that nilmanifold). By default, when we talk about equidistribution in a nilmanifold, we mean with respect to the Haar measure; thus is asymptotically equidistributed if and only if
for all Lipschitz . One can also describe single-scale equidistribution (and non-standard equidistribution) in a similar fashion, but for sake of discussion let us restrict attention to the simpler and more classical situation of asymptotic equidistribution here (although it is the single-scale equidistribution theory which is ultimately relevant to questions relating to the Gowers norms).
When studying equidistribution of polynomial sequences in a torus , a key tool was the van der Corput lemma. This lemma asserts that if a sequence is such that all derivatives with are asymptotically equidistributed, then itself is also asymptotically equidistributed.
The notion of a derivative requires the ability to perform subtraction on the range space : . When working in a higher degree nilmanifold , which is not a torus, we do not have a notion of subtraction. However, such manifolds are still torus bundles with torus . This gives a weaker notion of subtraction, namely the map , where is the diagonal action of the torus on the product space . This leads to a generalisation of the van der Corput lemma:
Lemma 10 (Relative van der Corput lemma) Let be a sequence in a degree nilmanifold for some . Suppose that the projection of to the degree filtered nilmanifold is asymptotically equidistributed, and suppose also that for each non-zero , the sequence is asymptotically equidistributed with respect to some -invariant measure on . Then is asymptotically equidistributed in .
Proof: It suffices to show that, for each Lipschitz function , that
By Exercise 20, we may assume that has a vertical frequency. If this vertical frequency is non-zero, then descends to a function on the degree filtered nilmanifold , and the claim then follows from the equidistribution hypothesis on this space. So suppose instead that has a non-zero vertical frequency. By vertically rotating (and using the -invariance of we conclude that . Applying the van der Corput inequality (see Notes 1), we now see that it suffices to show that
for each non-zero . The function on is -invariant (because of the vertical frequency hypothesis) and so descends to a function on . We thus have
The function has a non-zero vertical frequency with respect to the residual action of (or more precisely, of , which is isomorphic to ). As is invariant with respect to this action, the integral thus vanishes, as required.
This gives a useful criterion for equidistribution of polynomial orbits. Define a horizontal character to be a continuous homomorphism from to that annihilates (or equivalently, an element of the Pontryagin dual of the horizontal torus ). This is easily seen to be a torus. Let be the projection map.
Theorem 11 (Leibman equidistribution criterion) Let be a polynomial orbit on a degree filtered nilmanifold . Suppose that . Then is asymptotically equidistributed in if and only if is non-constant for each non-trivial horizontal character.
This theorem was first established by Leibman (by a slightly different method), and also follows from the above van der Corput lemma and some tedious additional computations; see this paper of Green and myself for details. For linear orbits, this result was established by Parry and by Leon Green. Using this criterion (together with more quantitative analogues for single-scale equidistribution), one can develop Ratner-type decompositions that generalise those in (Notes 1). Again, the details are technical and I refer to my paper with Green for details. We give a special case of Theorem 11 as an exercise:
Exercise 22 Use Lemma 10 to show that if are two real numbers such that are linearly independent modulo over the integers, then the polynomial orbit
is asymptotically equidistributed in the Heisenberg nilmanifold ; note that this is a special case of Theorem 11. Conclude that the map is asymptotically equidistributed in the unit circle.
Unfortunately Lemma 10 is not strong enough to cover all cases of Theorem 11; in particular, if are independent but are not, then the hypotheses of Lemma 10 are not obeyed for any fixed non-zero , although they are in some sense asymptotically obeyed in the limit when is large. To obtain Theorem 11 in this case one either needs a quantitative (single-scale) version of Lemma 10, or else one has to invoke the ergodic theorem in a number of places. The former approach is the one taken in the above mentioned paper of Green and myself, and the latter in the paper of Leibman.
One application of this equidistribution theory is to show that bracket polynomial objects such as (2) have a negligible correlation with any genuinely quadratic phase (or more generally, with any genuinely polynomial phase of bounded degree); this result was first established by Haland. On the other hand, from Theorem 7 we know that (2) has a large norm. This shows that even when , one cannot invert the Gowers norm purely using polynomial phases. This observation first appeared in the work of Gowers (with a related observation due to Furstenberg and Weiss).
Exercise 23 Let the notation be as in Exercise 22. Show that