On the relation between upper central quotients and lower central series of a group

Graham Ellis

Max-Planck-Institut Für Mathematik, Gottfried-Claren-Strasse 26, Bonn. Mathematics Department, National University of Ireland, Galway. (e-mail: graham.ellis@ucg.ie)

January 1999

1. Introduction A group H gives rise to an upper central series 1 = Z_{0}H ≤ Z_{1}H ≤ ⋯ and a lower central series H = γ_{1}H ≥ γ_{2}H ≥ · · ·. In this article we consider a normal subgroup N ⊴ H contained in Z_{c}H, and study the influence of the quotient G = H/N on the lower central group γ_{c+1}H. An old result of R. Baer [1][18] states that γ_{c+1}H is finite whenever the quotient G is finite. We develop Baer’s techniques to obtain the following three results.

A. For any finite G we give an upper bound on the order of γ_{c+1}H. (A previous paper [9] gave a bound on |γ_{c+1}H| when G is finite nilpotent; the present result incorporates a small improvement in this case. Several authors have given bounds when c = 1. In particular, there are papers by J.A. Green [17], J. Wiegold [29] [30], W. Gaschütz et. al. [15], and M.R. Jones [19] [20] [21]. The case c = 1 is also studied in [11] where the results obtained are slightly sharper than that got by specialising our general bound to c = 1.)

B. For any finite G we give an upper bound on the exponent of γ_{c+1}H. (For c = 1 this provides a generalisation of a result of A. Lubotzky and A. Mann [23] on the exponent of the Schur multiplier M(G) of a powerful p-group G; it also sharpens a bound of M.R. Jones [21] on the exponent of the Schur multiplier of a prime-power group. Furthermore, for c ≥ 1 our bound yields a generalisation and sharpening of an estimate, given in [7], on the exponent of the c-nilpotent Baer invariant M^{(c)}(G). This improvement for c ≥ 1 has the following practical implication. An electronically down-loadable appendix to the paper [12] contains a magma computer program for calculating a number of homotopy-theoretic constructions. In particular, it contains a function for computing M^{(c)}(G) which requires, as input data, a finite presentation of a finite group G together with any positive integer q divisible by e^{c} where e denotes the exponent of M^{(c)}(G). The improved estimate for e helps in choosing a suitable value for q.)

C. For G equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as γ_{c+1}H. (This extends work of N.D. Gupta and M.R.R. Moghaddam [16] which handles the dihedral 2-groups. It also extends work of D. MacHale and P.Ó’Murchú [26], and J. Burns et. al. [8] which treats all groups G of order at most 30 for c = 1, and all groups G of order at most 16 for c = 2.)

A precise statement of results A-C is provided in Section 2. Their proofs are given in Sections 4-6 respectively. The proofs involve three techniques with which the reader may not be too familiar. The first is the use of a nonabelian tensor product of groups. The second is the use of a Schur multiplier M(N, G) of a group G relative to a normal subgroup N. The third is the use of Baer invariants of a group. Relevant details of these techniques are recalled, and developed, in Section 3. Some results in Section 3 (in particular Propositions 5, 8 and 9) may be of independent interest.

2. Statement of results Let a group G be presented as the quotient of a free group F by a normal subgroup R. We state our results in terms of the Baer invariants

M^{(c)}(G) = \frac{R ∩ γ_{c+1}F}{γ_{c+1}(R, F)} , c ≥ 1,

and related invariants

γ_{c+1}^{∗}(G) = \frac{γ_{c+1}F}{γ_{c+1}(R, F)}

of the group G, where γ_{1}(R, F) = R, γ_{c+1}(R, F) = [γ_{c}(R, F), F] and γ_{c+1}F = γ_{c+1}(F, F). It was shown by R. Baer [1] (see also [14] [25]) that these invariants are, up to group isomorphism, independent of the choice of free presentation of G. Note that there are canonical actions of G on M^{(c)}(G) and γ_{c+1}^{∗}(G) given by conjugation in F/γ_{c+1}(R, F).

If G is of the form G ≅ H/N with N a normal subgroup of H contained in Z_{c}H, then it is routine [6] to establish the existence of the canonical short exact sequence

[figure: A; γ_{c+1}^{∗}(G); γ_{c+1}H]

where A is a submodule of the ZG-module M^{(c)}(G). We thus have inequalities (in which ≤ can be taken to mean ‘divides’)

|γ_{c+1}H| ≤ |γ_{c+1}^{∗}(G)| = |M^{(c)}(G)||γ_{c+1}G| ,  (1)
exp(γ_{c+1}H) ≤ exp(γ_{c+1}^{∗}(G)) ,  (2)

involving the orders and exponents of groups. Since M^{(c)}(G) is a subgroup of γ_{c+1}^{∗}(G), we also have an inequality

exp(M^{(c)}(G)) ≤ exp(γ_{c+1}^{∗}(G)) .  (3)

Furthermore, a group K arises as γ_{c+1}H for some H if and only if

K ≅ \frac{γ_{c+1}^{∗}(G)}{A}  (4)

with A a submodule of M^{(c)}(G).

Observations (1), (2) and (4) allow us to state results A-C in terms of the invariant γ_{c+1}^{∗}(G). For the statement of result A we let χ_{c}(d) denote the number of elements in a basis of the free abelian group γ_{c}F/γ_{c+1}F with F the free group on d generators. (There is a well-known formula for χ_{c}(d) due to E. Witt [27]. Let μ(m) be the Möbius function, defined for all positive integers by μ(1) = 1, μ(p) = −1 if p is a prime, μ(p^{k}) = 0 for k > 1, and μ(ab) = μ(a)μ(b) if a and b are coprime integers. Witt’s formula is

χ_{c}(d) = (1/c) ∑_{m|c} μ(m)d^{(c/m)}

where m runs through all divisors of c. Thus, for instance, χ_{2}(d) = (d^{2} − d)/2, χ_{3}(d) = (d^{3} − d)/3, χ_{4}(d) = (d^{4} − d^{2})/4.)

For an arbitrary finite abelian p-group A we define the integer

Λ_{c}(A) = e_{1}χ_{c+1}(d_{1}) + ∑_{j=2}^{k} e_{j}{χ_{c+1}(d_{1} + ⋯ + d_{j}) − χ_{c+1}(d_{1} + ⋯ + d_{j−1})}

where the parameters d_{j}, e_{j}, k are determined by expressing A uniquely in the form

A ≅ (C_{p^{e_{1}}} )^{d_{1}} × (C_{p^{e_{2}}} )^{d_{2}} × ⋯ × (C_{p^{e_{f}}})^{d_{k}}

with e_{1} > e_{2} > ⋯ > e_{k} > 1.

For an arbitrary finite d-generator p-group P we define the integer

Ψ_{c}(P) = m_{c}d + m_{c−1}d^{2} + ⋯ + m_{1}d^{c}

where the terms of the lower central series of P have orders |γ_{j}(P)| = p^{m_{j}}.

Note that an arbitrary finite group G has a smallest term L in its lower central series, namely the unique group L = γ_{r}G that satisfies γ_{r}G = γ_{r+1}G. Suppose that P is a d-generator p-Sylow subgroup of G with Frattini subgroup Φ(P) = [P, P]P^{p}, that P/(P ∩[G, G]) is a δ-generator group, that (L ∩ P)/(L ∩ Φ(P)) has order p^{t}, that L ∩ P has order p^{β}, and that [L ∩ P, P] has order p^{β′}. We use these various parameters to define the integer

Θ_{c}(G, L, P) = β + (ω − β′)(1 + δ + δ^{2} + ⋯ + δ^{c−1}) ,

where

ω = dβ − (1/2)t(t + 1) .

Theorem A. Let G be a finite group whose order has prime factors p_{1}, p_{2}, ⋯, p_{n}. Let L be the smallest term in the lower central series of G. The quotient G/L is nilpotent and thus a direct product

G/L ≅ S_{1} × S_{2} × ⋯ × S_{n}

with S_{i} a (possibly trivial) p_{i}-group. For each i let P_{i} be some p_{i}-Sylow subgroup of G. Then

|γ_{c+1}^{∗}(G)| ≤ ∏_{i=1}^{n} p_{i}^{Λ_{c}(S_{i}^{ab})+Ψ_{c}(S_{i})+Θ_{c}(G,L,P_{i})} .

The bound is attained, for instance, when G is abelian.

Note that if G is perfect then Λ_{c}(S_{i}^{ab}) = 0, Ψ_{c}(S_{i}) = 0 and Θ_{c}(G, L, P_{i}) = d_{i}(2β_{i} − d_{i} − 1)/2 + α_{i} where p_{i}^{β_{i}} is the order of a d_{i}-generator p_{i}-Sylow subgroup P_{i}, and p_{i}^{α_{i}} is the order of the abelianisation P_{i}^{ab}. If, at the other extreme, G is nilpotent then we have Θ_{c}(G, L, P_{i}) = 0. If G is abelian then Θ_{c}(G, L, P_{i}) = 0 and Ψ_{c}(S_{i}) = 0.

The bound in Theorem A can be sharpened by involving the relative Schur multiplier M(L, G) whose definition is recalled in Section 3. More precisely, in the definition of Θ_{c}(G, L, P) we can re-define ω = μ + β′ where the pth primary component of the abelian group M(L, G) has order |M(L, G)_{p}| = p^{μ}. For example, if |L| is coprime to |G|/ exp(L), then the relative multiplier is trivial (see Proposition 7(ii)) and we can take Θ_{c}(G, L, P_{i}) = β_{i} for c ≥ 1.

Before stating result B let us recall that A. Lubotzky and A. Mann [23] defined a p-group P to be powerful if: p ≥ 3 and [P, P] ⊂ P^{p}; or p = 2 and [P, P] ⊂ P^{4} (where P^{i} is the subgroup of P generated by all ith powers). In other words, P is powerful if p ≥ 3 and P/P^{p} is abelian, or if p = 2 and P/P^{4} is abelian. They proved a number of results about powerful groups P, one of which states that the exponent exp(M^{(1)}(P)) of the Schur multiplier divides the exponent of P. We shall generalise this. Our generalisation implies, for instance, that exp(M^{(c)}(P)) divides exp(P) for all c ≥ 1 and all P in a certain class C_{p} of p-groups; the class C_{p} consists of those p-groups P satisfying [[P^{p^{i−1}}, P], P] ⊂ P^{p^{i}} for 1 ≤ i ≤ e where exp(P) = p^{e}. It is shown in [23] that if P is powerful then [P^{p^{i−1}}, P] ⊂ P^{p^{i}}. Hence the class C_{p} contains all powerful p-groups.

Given a normal subgroup N ⊴ G of some group G, we say that the pair (N, G) has nilpotency class k if γ_{k+1}(N, G) = 1 and γ_{k}(N, G) ≠ 1. For a real number r we let [r] denote the smallest integer n such that n ≥ r.

Let N be a normal subgroup of a finite p-group P and suppose that N has exponent p^{e}. We define the integer

Ω(N, P) = [k_{1}/2] + [k_{2}/2] + ⋯ + [k_{e}/2]

where k_{j} denotes the nilpotency class of the pair (N^{p^{j−1}}/N^{p^{j}}, P/N^{p^{j}}) for 1 ≤ j ≤ e. For N equal to the trivial group we set Ω(1, P) = 0. Note that Ω(N, P) ≤ [k/2]e where k is the nilpotency class of P.

Theorem B. (i) Let G be a finite group whose order has prime factors p_{1}, p_{2}, ⋯, p_{n}. Let L be the smallest term in the lower central series of G. The quotient G/L is thus a direct product

G/L ≅ S_{1} × S_{2} × ⋯ × S_{n}

with S_{i} a (possibly trivial) p_{i}-group. For each i let P_{i} be a p_{i}-Sylow subgroup of G. Suppose that L ∩ P_{i}/[L ∩ P_{i}, P_{i}] has exponent p_{i}^{n_{i}}. Suppose that the p_{i}-primary component of G^{ab} has exponent p^{e_{i}} with e_{i} ≥ 0, and set m_{i} = min(Ω(L ∩ P_{i}, P_{i}), e_{i}). Then, for each c ≥ 1,

exp(γ_{c+1}^{∗}(G)) divides ∏_{i=1}^{n} p_{i}^{Ω(L∩P_{i},P_{i})+Ω(S_{i},S_{i})+n_{i}+(c−1)m_{i}} .

The bound is attained if G is abelian. (ii) Suppose that a p-group P satisfies [[P^{p^{i−1}}, P], P] ⊂ P^{p^{i}} for all 1 ≤ i ≤ e where p^{e} is the exponent of P. Then Ω(P, P) = e.

Note that, by inequality (3), exp(M^{(c)}(G)) divides exp(γ_{c+1}^{∗}(G)) for any group G. Thus, for an arbitrary finite p-group P of class k and exponent p^{e}, Theorem B(i) implies that exp(M^{(c)}(P)) divides p^{[k/2]e}; this sharpens the bound exp(M^{(c)}(P)) ≤ p^{(k−1)e} of Corollary 2.6 in [21] (for c = 1) and Theorem 6 in [7] (for c ≥ 1). Theorem B(ii) implies that exp(γ_{c+1}^{∗}(P)) divides exp(P) if, for example, P is a p-group with P/P^{p} of nilpotency class 2 and P^{p} contained in the second centre Z_{2}(P).

The bound in Theorem B(i) can be sharpened by re-defining m_{i} to be m_{i} = min(ϵ_{i}, e_{i}) where p_{i}^{ϵ_{i}} and p^{e_{i}} are the exponents of the p_{i}-primary components of M(L, G) and G^{ab} respectively (cf. Proposition 7 in Section 3). The bound is clearly independent of c if G is finite nilpotent, or if G is perfect. We do not know whether the bound can be made independent of c for arbitrary finite groups.

For the statement of result C we let D_{n} =< a, b | a^{2} = b^{n} = (ab)^{2} = 1 > denote the dihedral group of order 2m, and Q_{n} =< a, b | a^{2} = b^{n} = (ab)^{2} > denote the quaternion group of order 4n. Recall [3] that a p-group E is said to be extra-special if its commutator subgroup [E, E], its Frattini subgroup Φ(E), and its centre Z_{1}(E) coincide and have order p. The extra-special groups have order p^{2k+1} for k ≥ 1, with precisely two extra-special p-groups for each k (see [3]). We let E(p, k) denote an arbitrary extra-special p-group of order p^{2k+1}; we let E(p, k)^{+} and E(p, k)^{−} denote the extra-special p-groups of order p^{2k+1} and exponents p and p^{2} respectively. For c ≥ 1 we have the following.

Theorem C. (i) For each n ≥ 2 we have

γ_{c+1}^{∗}(D_{n}) ≅ \{ \begin{matrix}C_{n} & odd n, \\ C_{n} × (C_{2})^{χ_{c+1}(2)−1} & even n. \end{matrix}

The generator b ∈ D_{n} acts trivially on γ_{c+1}^{∗}(D_{n}); the generator a ∈ D_{n} acts trivially on elements of order two, and inverts the elements of the cyclic summand C_{n}. (ii) For each n ≥ 2 we have

γ_{c+1}^{∗}(Q_{n}) ≅ γ_{c+1}^{∗}(D_{n}) .

The generators a, b ∈ Q_{n} act as in (i). (iii) For each k ≥ 2 we have

γ_{c+1}^{∗}(E(p, k)) ≅ (C_{p})^{χ_{c+1}(2k)}.

The group E(p, k) acts trivially on γ_{c+1}^{∗}(E(p, k)). (iii)′ For p ≥ 3 and some 1 ≤ r ≤ 2^{c} we have

γ_{c+1}^{∗}(E(p, 1)^{+}) ≅ (C_{p})^{χ_{c+1}(2)+r},

γ_{c+1}^{∗}(E(p, 1)^{−}) ≅ (C_{p})^{χ_{c+1}(2)}.

Note that the corresponding Baer invariants M^{(c)}(G) are easily obtained applying the formula M^{(c)}(G) = ker(γ_{c+1}^{∗}(G) → γ_{c+1}(G)) to the precise details of the isomorphisms given in the proof of Theorem C. This extends the computations on dihedral 2-groups given in [16]. (We remark that there is a slip in the statement of the main theorem in [16]; the statement is correct for γ_{c+1}^{∗}(D_{2^{n}}) but incorrect for M^{(c)}(D_{2^{n}}).)

The precise value of r in Theorem C(iii)′ needs further investigation. The computer program listed in [12] yields the following results for the Burnside group B(2, 3) = E(3, 1)^{+} of exponent 3 on two generators.

\underline{c γ_{c+1}^{∗}(B(2, 3)) r}

\begin{matrix}1 & (C_{3})^{3} & 2 \\ 2 & (C_{3})^{5} & 3 \\ 3 & (C_{3})^{9} & 6 \\ 4 & (C_{3})^{15} & 9 \\ 5 & (C_{3})^{27} & 18 \end{matrix}

3. Preliminaries The tensor product of nonabelian groups is a convenient setting for performing commutator calculations. Its functorial properties make it especially suited to the task of relating commutator calculations in a group to those in a homomorphic image of the group. We begin this section by recalling and developing relevant details on this tensor product. We then recall details on a Schur multiplier M(N, G) defined for pairs of groups. By a pair of groups (N, G) we simply mean a group G with normal subgroup N. The advantage of working with pairs is that any finite group G can be expressed as an extension

[figure: (L, G); (G, G); (G/L, G/L)]

of a ‘perfect’ pair (L, G) by a ‘nilpotent’ pair (G/L, G/L). Various simplifications apply when dealing with the Schur multiplier of perfect or nilpotent pairs. We end the section with some details on Baer invariants.

Suppose given two groups G and H which act on each other via group actions G × H → H, (g, h) ↦ ^{g}h and H × G → G, (h, g) ↦ ^{h}g. Furthermore, suppose that each group acts on itself by conjugation, ^{x}y = xyx^{−1}. (In keeping with this notation, our convention for commutators is [x, y] = xyx^{−1}y^{−1}.) The tensor product G ⊗ H is defined [5] [4] to be the group generated by symbols g ⊗ h subject to the relations

gg′ ⊗ h = (^{g}g′ ⊗^{g}h) (g ⊗ h) ,
g ⊗ hh′ = (g ⊗ h) (^{h}g ⊗^{h}h′) ,

for g, g′ ∈ G, h, h′ ∈ H. The actions are said to be compatible if

(^{g}h)g′ = ^{g}(^{h}(^{g^{−1}}g′)) , ^{(^{h}g)}h′ = ^{h}(^{g}(^{h^{−1}}h′))

for all g, g′ ∈ G, h, h′ ∈ H.

Proposition 1. [5] Suppose that G and H act compatibly on each other.

(i) For all g, g′ ∈ G, h, h′ ∈ H the following identities hold in G ⊗ H:

g′ ⊗ (^{g}h) h^{−1} = ^{g′}(g ⊗ h) (g ⊗ h)^{−1} ,
g (^{h}g^{−1}) ⊗ h′ = (g ⊗ h)^{h′}(g ⊗ h)^{−1} ,
g (^{h}g^{−1}) ⊗ (^{g′}h′) h^{′−1} = [g ⊗ h, g′ ⊗ h′] .

(ii) There is a homomorphism ∂_{G}: G ⊗ H → G, g ⊗ h ↦ g^{h}g^{−1}. (iii) There is a ‘diagonal’ action of G on G ⊗ H given by ^{g′}(g ⊗ h) = (^{g′}g ⊗^{g′}h). (iv) There is an isomorphism G ⊗ H →^{≅} H ⊗ G, g ⊗ h ↦ h ⊗ g.

(v) If ^{g}h = h, ^{h}g = g for all g ∈ G, h ∈ H then G ⊗ H ≅ G^{ab} ⊗ H^{ab}, where the right-hand side of the isomorphism denotes the usual tensor product of abelian groups.

For each pair of groups (N, G) we can form the tensor product N ⊗ G in which all actions are taken to be conjugation in G. Since conjugation yields compatible actions there is a diagonal action of G on N ⊗ G. The tensor product N ⊗ G acts on G by conjugation in G via the homomorphism ∂_{N}: N ⊗ G → N. We can thus construct the triple tensor product (N ⊗ G) ⊗ G. One readily checks that the construction preserves ‘compatibility of actions’, and that it can therefore be iterated to form the (c + 1)-fold tensor product

⊗^{c+1}(N, G) = (⋯ ((N ⊗ G) ⊗ G) ⊗ ⋯ ⊗ G) c ≥ 1,

involving c copies of G and one copy of N.

Proposition 2. [13] Let G be a d-generator p-group with normal subgroup N. Suppose that γ_{i}(N, G) = p^{m_{i}} for i ≥ 1, m_{i} ≥ 0. Then, for any c ≥ 1, we have

| ⊗^{c+1} (N, G)| ≤ p^{m_{c}d+m_{c−1}d^{2}+···+m_{1}d^{c}} .

Lemma 3. Let G_{3} ↪ G_{2}↠G_{1}, H_{3} ↪ H_{2}↠H_{1} be two short exact sequences of groups. Suppose that G_{i} and H_{i} act compatibly on one another for 1 ≤ i ≤ 3, and that the homomorphisms preserve actions. Then there is an exact sequence of homomorphisms

(G_{3} ⊗ H_{2})\overline{×}(G_{2} ⊗ H_{3}) ⟶ G_{2} ⊗ H_{2} ⟶ G_{1} ⊗ H_{1} ⟶ 1

in which \overline{×} denotes a semi-direct product whose details need not be specified.

Proof. The lemma is a routine adaption of Proposition 9 in [4]. □

Lemma 4. Let N be a normal subgroup of G for which the commutator [n, [n, g]] is trivial for all g ∈ G, n ∈ N. In the tensor product N ⊗ G, with G and H acting by conjugation, the following identity holds for all g ∈ G, n ∈ N and all integers t ≥ 2:

n^{t} ⊗ g = (n ⊗ g)^{t} (n ⊗ [n, g]^{t(t−1)/2}) .

Proof. The case N = G is proved in [2]. The proof of the more general case is analogous; it can also be derived directly using Proposition 1(i). □

Recall that a pair (N, G) is said to be nilpotent of class k if γ_{k+1}(N, G) = 1 and γ_{k}(N, G) ≠ 1. Also recall that [k/2] denotes the smallest integer n such that n ≥ k/2.

Proposition 5. Let G be a group with normal subgroup N. Suppose that N has prime-power exponent p^{e} and that the pair (N, G) has nilpotency class ≤ k. Then, for any c ≥ 1, we have

exp(⊗^{c+1}(N, G)) divides p^{[k/2]e}.

Proof. For t = p^{e} the binomial coefficient \binom{t}{2} is divisible by t when p ≥ 3, and divisible by t/2 when p = 2. Thus Lemma 4 proves the proposition for k = 2, c = 1 ( since for p = 2, t = p^{e} and γ_{3}(N, G) = 1 the identity

n ⊗ [n, g]^{t/2} = n ⊗ [n^{t/2}, g]

holds for all g ∈ G, n ∈ N; but [n^{t/2}, g] = 1 because n^{t/2} has order at most 2.)

Let us now consider k = 2 and some c ≥ 2. Then

⊗^{c+1}(N, G) = ⊗^{c}(N ⊗ G, G)

and N ⊗ G acts trivially on G. The triviality of this action implies the identity

(⋯ ((n ⊗ g)^{t} ⊗ g_{1}) ⊗ ⋯ ⊗ g_{c}) = (⋯ ((n ⊗ g) ⊗ g_{1}) ⊗ ⋯ ⊗ g_{c})^{t}

in ⊗^{c+1}(N, G). Hence exp(⊗^{c+1}(N, G)) divides exp(N ⊗ G) and the proposition is proved for k = 2, c ≥ 1.

Suppose now that the proposition has been proved for some c and all k < k_{0}. Suppose γ_{k_{0}+1}(N, G) = 1. Lemma 3 implies an exact sequence

(γ_{k_{0}−1}(N, G) ⊗ G)\overline{×}(N ⊗ γ_{k_{0}−1}(N, G)) → N ⊗ G → \frac{N}{γ_{k_{0}−1}(N, G)} ⊗ \frac{G}{γ_{k_{0}−1}(N, G)} .

Working in N ⊗ G, the image of γ_{k_{0}−1}(N, G) ⊗ G contains the image of N ⊗ γ_{k_{0}−1}(N, G) by virtue of the identity

m ⊗ [n, g] = ([n, g] ⊗ m)^{−1}

which follows from Proposition 1(i) for all g ∈ G, m, n ∈ N. We thus have an exact sequence

γ_{k_{0}−1}(N, G) ⊗ G → N ⊗ G → \frac{N}{γ_{k_{0}−1}(N, G)} ⊗ \frac{G}{γ_{k_{0}−1}(N, G)} .

By applying Lemma 3 to this sequence, and invoking a similar identity, we obtain the exact sequence

(γ_{k_{0}−1}(N, G) ⊗ G) ⊗ G → (N ⊗ G) ⊗ G → (\frac{N}{γ_{k_{0}−1}(N, G)} ⊗ \frac{G}{γ_{k_{0}−1}(N, G)}) ⊗ \frac{G}{γ_{k_{0}−1}(N, G)} .

Repetition of the process yields an exact sequence

⊗^{c+1}(γ_{k_{0}−1}(N, G), G) → ⊗^{c+1}(N, G) → ⊗^{c+1}(\frac{N}{γ_{k_{0}−1}(N, G)}, \frac{G}{γ_{k_{0}−1}(N, G)})

from which we deduce that exp(⊗^{c+1}(N, G)) ≤ p^{[(k_{0}−2)/2]}p^{e} = p^{[k_{0}/2]}. By induction, the proposition is proved for all c, k ≥ 1. □

Following J.-L.Loday [22] we say that a pair of groups (N, G) is perfect if N = [N, G].

Proposition 6. Let (N, G) be any perfect pair of groups and set M = ker(∂_{N}: N ⊗G → N). Then M is abelian and, for each c ≥ 1, there is an exact sequence

⊗^{c+1}(M, G^{ab}) → ⊗^{c+2}(N, G) → ⊗^{c+1}(N, G) → 1

where ⊗^{c+1}(M, G^{ab}) is the usual iterated tensor product of abelian groups.

Proof. Let H_{3} ↪ H_{2}↠H_{1} be a short exact sequence of groups, and let G be a group such that G and H_{i} act compatibly on each other for 1 ≤ i ≤ 3 with the homomorphisms preserving the actions. Then H_{3} acts trivially on G via H_{2}. Suppose that the action of G on H_{2} restricts to a trivial action of G on H_{3}. Then Proposition 1(v) and Lemma 3 imply an exact sequence

H_{3}^{ab} ⊗ G^{ab} → H_{2} ⊗ G → H_{1} ⊗ G → 1.  (5)

A perfect pair of groups (N, G) gives rise to a short exact sequence ker(∂_{N}) ↪ N ⊗ G ↠^{∂} N. The identity

h(n ⊗ g) = ([n, g] ⊗ h)^{−1} (n ⊗ g)

which holds in N ⊗ G (see Proposition 1) for all g, h ∈ G, n ∈ N implies that G acts trivially on ker(∂_{N}). So (5) implies an exact sequence

M ⊗ G^{ab}→(N ⊗ G) ⊗ G ⟶^{∂⊗1} N ⊗ G → 1 .

Note that the diagonal action of G on M ⊗ G^{ab} is trivial, and hence G acts trivially on ker(∂ ⊗ 1). Thus a second application of (5) yields the exact sequence ker(∂ ⊗ 1) ⊗ G^{ab} → ⊗^{4}(N, G) → ⊗^{3}(N, G) → 1. From this we derive the exact sequence ⊗^{3}(M, G^{ab}) → ⊗^{4}(N, G) → ⊗^{3}(N, G) → 1. The proposition follows from a repetition of this argument. □

Given a pair of groups (N, G) we denote by Δ(N, G) the subgroup of N ⊗ G generated by the elements n ⊗ n for n ∈ N. This is a normal subgroup and following [5] we define the exterior product

N ∧ G = N ⊗ G/Δ(N, G) .

The homomorphism ∂_{N}: N ⊗ G → N clearly induces a homomorphism ∂_{N}: N ∧ G → N. The identity

[g, n] ⊗ [g′, n′] = [(g ⊗ n), (g′ ⊗ n′)] ,

of Proposition 1(i) implies an isomorphism N ∧ G ≅ N ⊗ G in the case of perfect pairs.

Definition. [10] The Schur multiplier of a pair of groups (N, G) is the group M(N, G) defined by

M(N, G) = ker(∂_{N}: N ∧ G → N) .

If the pair is perfect then, equivalently,

M(N, G) = ker(∂_{N}: N ⊗ G → N) .

Proposition 7. [10] Let G be a finite group with normal subgroup N ⊴ G.

(i) Then M(N, G) is a finite abelian group with exponent e dividing the order of G. (ii) Let e′ denote the exponent of N. Then, in fact, ee′ divides the order of G and e divides the order of N. (iii) Let K be any subgroup of G such that each g ∈ G can be expressed (not necessarily uniquely) as a product g = nk with n ∈ N, k ∈ K. Then e^{2} divides |N| × |K|. (iv) Suppose that P is a p-Sylow subgroup of G. Let M(P ∩ N, P)_{p} denote the p-component of the multiplier, and ι: M(P ∩ N, P) → M(N, G) the homomorphism induced by inclusion. Then

M(P ∩ N, P) ≅ M(N, G)_{p} ⊕ ker(ι) .

Proposition 8. Let G be a δ-generator p-group with normal subgroup N ⊴ G of order |N| = p^{β}. Suppose that |N/(N ∩ Φ(G))| = p^{t} where Φ(G) = [G, G]G^{p}. Then

|N ∧ G| ≤ p^{δβ− \frac{t(t+1)}{2}} .

Proof. Proposition 2 implies that |N ⊗ G| ≤ p^{δβ}. There is a commutative diagram of group homomorphisms

\xymatrix{ & Γ(\frac{N}{[N,G]}) \ar[r]^{Δ′} \ar[dd] & N ⊗ G \ar[r] \ar[d] & N ∧ G \ar[r] & 1 \\  &  & G ⊗ G \ar@{->>}[d] \\ 1 \ar[r] & Γ(\frac{G}{Φ(G)}) \ar[r]^{Δ} & \frac{G}{Φ(G)} ⊗ \frac{G}{Φ(G)} \ar[r] & \frac{G}{Φ(G)} ∧ \frac{G}{Φ(G)} \ar[r] & 1}

in which the rows (but not the columns) are exact [5]. The abelian group Γ(A), defined for any additive abelian group A, is J.H.C. Whitehead’s universal quadratic construction; it is generated (as an abelian group) by symbols γ(a) for a ∈ A subject to the relations

γ(−a) = γ(a)
γ(a + b + c) + γ(a) + γ(b) + γ(c) = γ(a + b) + γ(a + c) + γ(b + c)

for a, b, c ∈ A. The homomorphism Δ is defined on generators by Δ(γ(x)) = x ⊗ x for x ∈ G/Φ(G). The image of Γ(G/[N, G]) in G/Φ(G) ⊗ G/Φ(G) is an elementary abelian group of rank t(t + 1)/2. Hence the exactness of the top row implies

|N ∧ G| = \frac{|N ⊗ G|}{|Δ′(Γ(N/[N, G]))|} ≤ \frac{p^{δβ}}{p^{t(t+1)/2}} .

This proves the proposition. □

Proposition 9. Let N be a normal subgroup of a group G. If N has exponent p^{e} then

exp(N ∧ G) divides p^{[k_{1}/2]+[k_{2}/2]+···+[k_{e}/2]}

where k_{i} denotes the nilpotency class of the pair (N^{p^{i−1}}/N^{p^{i}}, G/N^{p^{i}}) for 1 ≤ i ≤ e.

Proof. Let K, M be normal subgroups of G with K ≤ M. Using the identity m ⊗ k = (k ⊗ m)^{−1} which holds in M ∧ G for all k ∈ K, m ∈ M, one readily develops the short exact sequence

K ∧ G → M ∧ G → M/K ∧ G/K → 1  (6)

from Lemma 3. Now (6) yields the exact sequences

N^{p^{i}} ∧ G → N^{p^{i−1}} ∧ G → N^{p^{i−1}}/N^{p^{i}} ∧ G/N^{p^{i}} → 1

for i ≥ 1. Hence

exp(N ∧ G) ≤ ∏_{i=1}^{e} exp(N^{p^{i−1}}/N^{p^{i}} ∧ G/N^{p^{i}}) .

Proposition 5 implies exp(N^{p^{i−1}}/N^{p^{i}} ∧ G/N^{p^{i}}) ≤ p^{[k_{i}/2]}. □

Tensor products are related to Baer invariants by the following slight generalisation of a result of A.-S.T. Lue [24] (cf. [6]).

Proposition 10. [24] For any group G with normal subgroup N ⊴ G, and for c ≥ 1, there is an exact sequence

⊗^{c+1}(N, G) → γ_{c+1}^{∗}(G) → γ_{c+1}^{∗}(G/N) → 1 .

It is convenient to set

\overline{γ}_{c+1}^{∗}(N, G) = ker(γ_{c+1}^{∗}(G) → γ_{c+1}^{∗}(G/N)) .

Note that γ_{c+1}^{∗}(G) = \overline{γ}_{c+1}^{∗}(G, G). (The bar is intended to suggest that \overline{γ}_{c+1}^{∗}(N, G) is a quotient of some functor γ_{c+1}^{∗}(N, G). For example, we can take γ_{2}^{∗}(N, G) = N ∧ G [6]. Proposition 9 and the following Proposition 11 could be subsumed under a single result concerning γ_{c+1}^{∗}(N, G).)

Proposition 11. Suppose that N is a normal subgroup of a group G. If N has exponent p^{e} then

exp(\overline{γ}_{c+1}^{∗}(N, G)) divides p^{[k_{1}/2]+[k_{2}/2]+···+[k_{e}/2]}

where k_{i} denotes the nilpotency class of the pair (N^{p^{i−1}}/N^{p^{i}}, G/N^{p^{i}}) for 1 ≤ i ≤ e.

Proof. The proof is analogous to that of Proposition 9, but with (6) replaced by the exact sequence

\overline{γ}_{c+1}^{∗}(K, G) → \overline{γ}_{c+1}^{∗}(M, G) → \overline{γ}_{c+1}^{∗}(M/K, G/K) → 1 .

We leave the verification of the exactness of this (canonical) sequence to the reader. □

The upper epicentral series of an arbitrary group G was introduced in [6]. This is a family of characteristic subgroups 1 = Z_{0}^{∗}(G) ≤ Z_{1}^{∗}(G) ≤ Z_{2}^{∗}(G) ≤ ⋯ with various useful properties such as those listed in the next proposition. Part (i) of the following proposition can be taken as the definition of Z_{c}^{∗}(G).

Proposition 12. [6] Let c ≥ 1.

(i) Z_{c}^{∗}(G) is the smallest normal subgroup of G, contained in Z_{c}(G), such that the quotient G/Z_{c}^{∗}(G) is isomorphic to H/Z_{c}H for some group H. (ii) Z_{c+1}^{∗}(G) contains Z_{c}^{∗}(G). (iii) Z_{c}^{∗}(G) = 1 if and only if there exists an isomorphism G ≅ H/Z_{c}H for some group H. (iv) Let N be a normal subgroup of G. Then N ≤ Z_{c}^{∗}(G) if and only if the quotient homomorphism G↠G/N induces an isomorphism γ_{c+1}^{∗}(G) →^{≅} γ_{c+1}^{∗}(G/N).

Let A be a d-generator abelian group with generators a_{1}, ⋯, a_{d}. Let A_{i} denote the cyclic subgroup of A generated by a_{i}. Let L(d) denote the set of basic commutators on the d symbols a_{i}. To each basic commutator λ = [a_{i_{1}}, ⋯, a_{i_{k}}] of weight k we associate the k-fold tensor product of abelian groups T (λ) = A_{i_{1}} ⊗⋯⊗A_{i_{k}}. Thus T is a cyclic group of order equal to the highest common factor of the orders of the A_{i_{j}}. It is explained in [9] that the invariant γ_{c+1}^{∗}(A) is isomorphic to a direct sum of cyclic groups

γ_{c+1}^{∗}(A) ≅ ⊕_{λ∈L(d)}T (λ) .

The following proposition is an immediate corollary to this isomorphism. An alternative derivation of the proposition can be found in [28].

Proposition 13. Let A be a direct product of cyclic groups

A = (C_{n_{1}})^{d_{1}} × (C_{n_{2}})^{d_{2}} × ⋯ × (C_{n_{k}})^{d_{k}}

with each n_{i} divisible by n_{i+1}. Then

γ_{c+1}^{∗}(A) ≅ (C_{n_{1}})^{χ_{c+1}(d_{1})} × ∏_{j=2}^{k}(C_{n_{j}})^{{χ_{c+1}(d_{1}+···+d_{j})−χ_{c+1}(d_{1}+···+d_{j−1})}} .

Proposition 14. [9] Let G = S_{1} ×S_{2} ×⋯×S_{k} be a direct product of groups whose abelianisations S_{i}^{ab} have finite, and mutually coprime, orders. Then, for each c ≥ 1, there is an isomorphism

γ_{c+1}^{∗}(G) ≅ γ_{c+1}^{∗}(S_{1}) × ⋯ × γ_{c+1}^{∗}(S_{k}) .

Proposition 15. Let N be a non-trivial normal subgroup of a p-group G. Let K denote the kernel of the canonical surjection γ_{c+1}^{∗}(G)↠γ_{c+1}^{∗}(G/N).

(i) K is non-trivial if and only if there exists some group H for which H/Z_{c}H ≅ G. (ii) If p ≥ 3 and N ⊂ G^{p} ∩ Z_{2}G, or if p = 2 and N ⊂ G^{p^{2}} ∩ Z_{2}G, then K is contained in the Frattini subgroup of γ_{c+1}^{∗}(G). (iii) If N is a proper subgroup of a cyclic normal subgroup in G, and if N ⊂ Z_{2}G, then K is contained in the Frattini subgroup of γ_{c+1}^{∗}(G).

Proof. Proposition 12 implies (i).

Proposition 10 implies that K is generated by the image of tensors of the form (⋯ ((n ⊗ g_{1}) ⊗ g_{2})⊗⋯⊗g_{c}). The hypothesis of (ii) with Lemma 4 implies that the canonical image in ⊗^{c+1}(G, G) of each such tensor lies in the subgroup ⊗^{c+1}(G, G)^{p} generated by pth powers of tensors. The hypothesis of (iii) implies that the image lies in the subgroup generated by pth powers of tensors together with tensors of the form (⋯ ((g ⊗ g^{t}) ⊗ g_{2}) ⊗ ⋯ ⊗ g_{c}). In both cases K lies in the Frattini subgroup of the p-group γ_{c+1}^{∗}(G). □

4. Proof of Theorem A Let G, S_{i}, P_{i}, L be as in the statement of Theorem A. For each prime p_{i} let ⊗^{c+1}(L, G)_{p_{i}} denote some p_{i}-Sylow subgroup of ⊗^{c+1}(L, G), and set

Λ_{c}^{i} = log_{p_{i}} |γ_{c+1}^{∗}(S_{i}^{ab})| ,
Ψ_{c}^{i} = log_{p_{i}} | ⊗^{c+1} ([S_{i}, S_{i}], S_{i})| ,
Θ_{c}^{i} = log_{p_{i}} | ⊗^{c+1} (L, G)_{p_{i}}| .

Propositions 10 and 14 imply exact sequences

⊗^{c+1}(L, G) → γ_{c+1}^{∗}(G) → ∏_{i=1}^{n} γ_{c+1}^{∗}(S_{i}) → 1 ,

⊗^{c+1}([S_{i}, S_{i}], S_{i}) → γ_{c+1}^{∗}(S_{i}) → γ_{c+1}^{∗}(S_{i}^{ab}) → 1 .

Hence

|γ_{c+1}^{∗}(G)| ≤ ∏_{i=1}^{n} p_{i}^{Λ_{c}^{i}+Ψ_{c}^{i}+Θ_{c}^{i}} .

To complete the proof we must find appropriate upper bounds Λ_{c}(S_{i}^{ab}), Ψ_{c}(S_{i}), Θ_{c}(G, L, P_{i}) for Λ_{c}^{i}, Ψ_{c}^{i}, Θ_{c}^{i}.

Proposition 13 furnishes the appropriate formula for Λ_{c}(S_{i}^{ab}). Proposition 2 provides the appropriate formula for Ψ_{c}(S_{i}). Suppose that P_{i} is a d_{i}-generator group, that P_{i}/(P_{i} ∩ [G, G]) is a δ_{i}-generator group, that (L ∩ P_{i})/(L ∩ Φ(P_{i})) has order p^{t_{i}}, that L ∩ P_{i} has order p_{i}^{β_{i}}, and that [L ∩ P_{i}, P_{i}] has order p^{β′_{i}}. Set M = M(L, G) = ker(L ⊗ G↠L) and let M_{p_{i}} denote the p_{i}-primary component of M. Since the pair (L, G) is perfect, Proposition 6 implies

| ⊗^{c+1} (L, G)| ≤ | ⊗^{c} (M, G^{ab})| × | ⊗^{c−1} (M, G^{ab})| × ⋯ × | ⊗^{2} (M, G^{ab})| × |M| × |L| ,

and thus

| ⊗^{c+1} (L, G)_{p_{i}}| ≤ | ⊗^{c} (M_{p_{i}}, G^{ab})| × | ⊗^{c−1} (M_{p_{i}}, G^{ab})| × ⋯ × | ⊗^{2} (M_{p_{i}}, G^{ab})| × |M_{p_{i}}| × |L ∩ P_{i}| .

Proposition 7(iv) implies that M_{p_{i}} ⊂ M(L ∩ P_{i}, P_{i}). Since |L ∩ P_{i}| = p_{i}^{β_{i}}, Proposition 8 implies that

|M(L ∩ P_{i}, P_{i})| × |[L ∩ P_{i}, P_{i}]| ≤ p_{i}^{d_{i}β_{i}− \frac{t_{i}(t_{i}+1)}{2}} .

Setting ω_{i} = d_{i}β_{i} − (1/2)t_{i}(t_{i} + 1), we have

|M_{p_{i}}| ≤ |M(L ∩ P_{i}, P_{i})| ≤ p_{i}^{ω_{i}−β′_{i}}

and hence

| ⊗^{c+1} (L, G)_{p_{i}}| ≤ p_{i}^{(ω_{i}−β′_{i})δ_{i}^{c}}p_{i}^{(ω_{i}−β′_{i})δ_{i}^{(c−1)}} ⋯ p_{i}^{(ω_{i}−β′_{i})δ_{i}}p_{i}^{(ω_{i}−β′_{i})}p_{i}^{β_{i}} .

This yields the appropriate formula for Θ_{c}(G, L, P_{i}).

5. Proof of Theorem B Let G, S_{i}, P_{i}, L be as in the statement of Theorem B. Let \overline{γ}_{c+1}^{∗}(L, G)_{p_{i}} denote a p_{i}-Sylow subgroup of the group \overline{γ}_{c+1}^{∗}(L, G). The short exact sequence \overline{γ}_{c+1}^{∗}(L, G) ↪ γ_{c+1}^{∗}(G)↠γ_{c+1}^{∗}(G/L) with Proposition 14 implies

exp(γ_{c+1}^{∗}(G)) ≤ ∏_{i=1}^{n} exp(\overline{γ}_{c+1}^{∗}(L, G)_{p_{i}}) exp(γ_{c+1}^{∗}(S_{i})) .

We apply Propositions 6 and 10 and the surjection ⊗^{c+1}(L, G)↠\overline{γ}_{c+1}^{∗}(L, G) to obtain

exp(γ_{c+1}^{∗}(G)) ≤ ∏_{i=1}^{n} exp((M(L, G) ⊗ G^{ab})_{p_{i}})^{c−1} exp((L ⊗ G)_{p_{i}}) exp(γ_{c+1}^{∗}(S_{i})) .

Proposition 7(iv) yields

exp(γ_{c+1}^{∗}(G)) ≤ ∏_{i=1}^{n} exp(M(L ∩ P_{i}, P_{i}) ⊗ G^{ab})^{c−1} exp((L ⊗ G)_{p_{i}}) exp(γ_{c+1}^{∗}(S_{i})) .

The exact sequence

(L ∩ P_{i}) ∧ P_{i} → (L ⊗ G)_{p_{i}} → L ∩ P_{i}/[L ∩ P_{i}, P_{i}] → 1

is readily derived, and yields

exp(γ_{c+1}^{∗}(G)) ≤ ∏_{i=1}^{n} exp(M(L ∩ P_{i}, P_{i}) ⊗ G^{ab})^{c−1} exp((L ∩ P_{i}) ∧ P_{i}) exp(\frac{L ∩ P_{i}}{[L ∩ P_{i}, P_{i}]}) exp(γ_{c+1}^{∗}(S_{i})) .

The bound of Theorem B(i) now follows from Propositions 9 and 11. (We take N = G = S_{i} in Proposition 11.)

To prove Theorem B(ii) it suffices to note that the condition [[P^{p^{i−1}}, P], P] ⊂ P^{p^{i}} is equivalent to saying that the pair (P^{p^{i−1}}/P^{p^{i}}, P/P^{p^{i}}) has nilpotency class at most 2.

6. Proof of Theorem C Consider the dihedral group D_{n} =< a, b | a^{2} = b^{n} = (ab) > with n = 2^{r}m where m ≥ 1 is odd. The smallest term of the lower central series of D_{n} is L = γ_{r}(D_{n}) ≅ C_{m}. Proposition 7(ii) implies that the relative multiplier M(L, G) is trivial. Proposition 6 therefore implies an isomorphism ⊗^{c+1}(L, G) ≅ L ⊗ G for all c ≥ 1. So Proposition 10 yields a commutative diagram

[figure: M(L, D_{n}) = 1; M^{(c)}(D_{n}); ≅; M^{(c)}(D_{2^{r}}); L/γ_{c+1}(L, D_{n}) = 1; L ⊗ D_{n}; γ_{c+1}^{∗}(D_{n}); γ_{c+1}^{∗}(D_{2^{r}}); 1; 1; C_{m}; γ_{c+1}D_{n}; γ_{c+1}D_{2^{r}}; 1]

in which the rows are exact and the columns are short exact. From this, and the isomorphism γ_{c+1}D_{n} ≅ C_{m} × γ_{c+1}D_{2^{r}}, we derive the isomorphism

γ_{c+1}^{∗}(D_{n}) ≅ C_{m} × γ_{c+1}^{∗}(D_{2^{r}}) .

A description of γ_{c+1}^{∗}(D_{2^{r}}) is given in [16]. Alternatively, the description can be re-obtained as follows. The central extensions C_{2} ↪ D_{2^{r}}↠D_{2^{r−1}} and repeated applications of Proposition 15(i) imply that |γ_{c+1}^{∗}(D_{2^{r}})| ≥ |γ_{c+1}^{∗}(C_{2} × C_{2})| × 2^{r−1}. These central extensions together with repeated applications of Proposition 15(iii) imply that γ_{c+1}^{∗}(D_{2^{r}}) has the same number of generators as γ_{c+1}^{∗}(C_{2} ×C_{2}), namely one generator for each basic commutator on two generators a, b. Theorem B implies that exp(γ_{c+1}^{∗}(D_{2^{r}})) ≤ 2^{r}. So, to obtain the isomorphism γ_{c+1}^{∗}(D_{2^{r}}) ≅ C_{2^{r}} ⊕ (C_{2})^{χ_{c+1}(2)−1} it suffices to verify that at least all but one of the generators are of order 2, and that the generators a, b act as stated in the theorem. (Note that the invariant γ_{c+1}^{∗}(G) is abelian precisely when γ_{c+1}(G) acts trivially on it.) This verification, which we leave to the reader, yields the desired description of γ_{c+1}^{∗}(D_{2^{r}}) and completes the proof of part (i) of the theorem.

To prove part (ii) we note that the quaternion group Q_{n} is not of the form H/Z(H) for any group H with centre Z(H) [3]. Proposition 12(ii)(iii) thus implies that the cth term Z_{c}^{∗}(Q_{n}) of the upper epicentral series of Q_{n} contains the centre Z(Q_{n}) ≅ C_{2}. Since Q_{n}/Z(Q_{n}) ≅ D_{n}, the isomorphism γ_{c+1}^{∗}(Q_{n}) ≅ γ_{c+1}^{∗}(D_{n}) follows from Proposition 12(iv).

To prove part (iii) we note that an extraspecial group E(p, k), k ≥ 2, is not of the form H/Z(H) for any group H with centre Z(H) [3]. Arguing as in the previous paragraph, we see that γ_{c+1}^{∗}(E(p, k)) ≅ γ_{c+1}^{∗}(E(p, k)/Z(E(p, k))) ≅ γ_{c+1}^{∗}(C_{p} × C_{p}). Proposition 13 completes the proof of part (iii). This argument also holds for E(p, 1)^{−}, p ≥ 3.

To obtain our partial description of γ_{c+1}^{∗}(E(p, 1)^{+}), p ≥ 3, we first remark that Z_{1}^{∗}(G) is trivial for the group G = E(p, 1)^{+} [3]. Letting Z = Z(G) denote the centre of this group, Proposition 10 and Proposition 1(v) yield an exact sequence ⊗^{c+1}(Z, G) = Z ⊗ G^{ab} ⊗ ⋯ ⊗ G^{ab} → γ_{c+1}^{∗}(G) ⟶^{ϕ} γ_{c+1}^{∗}(C_{p} × C_{p}) → 1. The group ⊗^{c+1}(Z, G) is elementary abelian of rank 2^{c}, and Proposition 12(iv) implies that ϕ has non-trivial kernel. Theorem B implies that exp(γ_{c+1}^{∗}(G)) = p. The commutator subgroup [G, G] = Z acts trivially on γ_{c+1}^{∗}(G), and so γ_{c+1}^{∗}(G) is abelian. Hence γ_{c+1}^{∗}(G) is elementary abelian of rank at most χ_{c+1}(2) + 2^{c}, and at least χ_{c+1}(2) + 1.

Acknowledgements. This research was carried out at the Max Planck Institut für Mathematik, Bonn. I would like to thank the Institute for its very generous hospitality.

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