PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 9, September 1998, Pages 2513–2523 S 0002-9939(98)04440-2

A BOUND FOR THE DERIVED AND FRATTINI SUBGROUPS OF A PRIME-POWER GROUP

GRAHAM ELLIS

(Communicated by Ronald M. Solomon)

Abstract. This paper is based on the seemingly new observation that the Schur multiplier M(G) of a d-generator group of prime-power order p^{n} has order |M(G)| ≤ p^{d(2n−d−1)/2}. We prove several related results, including sufficient conditions for a sharper bound on |M(G)| to be an equality.

Let G be a group with centre Z. Schur [13] observed that the commutator subgroup [G, G] is finite whenever the quotient G/Z is finite. Wiegold [17] obtained an estimate for the order of [G, G] in terms of the order of G/Z; in particular he showed that if G/Z has prime-power order p^{n}, then the order of [G, G] is at most p^{n(n−1)/2}. This bound can be attained for all n ≥ 1, but only if G/Z is elementary abelian. As a corollary, Wiegold re-derived Green’s bound [7] on the order of the Schur multiplier of a prime-power group. Gaschütz, Neubüser and Yen [6] subsequently refined Wiegold’s methods to obtain a sharper bound on [G, G] (involving the number of generators of the quotient of G by its second centre), and a corresponding sharpening of Green’s bound. By a different fine tuning of Wiegold’s methods, Jones ([9], [10], [11]) obtained complementary improvements in Green’s bound.

In this article we reduce the above bounds on [G, G] by incorporating, in turn:

(1) the number of generators of G/Z; (2) the abelian group structure of the quotients of the lower central series of G/Z; (3) the restricted Lie algebra structure of the quotients of the lower p-central series of G/Z; (4) the “breadth in G” of the preimages of the generators of G/Z. The fourth approach (which is the only one to involve more than the structure of G/Z) complements a result of Vaughan-Lee [16]. We obtain corresponding reductions in the above-mentioned bounds on the Schur multiplier. As an application we obtain a rough upper bound on the number of d-generator groups of order p^{n}. Furthermore, our results are presented in such a way as to yield bounds on the Frattini subgroup of a prime-power group.

Let p be any prime, and let q ≥ 0 be any nonnegative integer multiple of p. (We are interested mainly in the cases q = 0 and q = p.) Given a group G we let Z^{q}(G) denote the subgroup of the centre of G consisting of those elements with order dividing q. Given a normal subgroup N in G, we let N#^{q}G denote the subgroup of G generated by the commutators ngn^{−1}g^{−1} and powers n^{q} for n ∈ N, g ∈ G.

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors January 27, 1997.

1991 Mathematics Subject Classification. Primary 20J05.

We define

γ_{1}^{q}G = G,
γ_{i+1}^{q}G = (γ_{i}^{q}G)#^{q}G,  i ≥ 1,
Γ_{i}^{q}G = γ_{i}^{q}G/γ_{i+1}^{q}G,  i ≥ 1.

Furthermore, we set

M^{q}(G) = H_{2}(G, ℤ_{q}),

H_{n}^{q}(G) = H_{n}(G, ℤ_{q}),

R(G, q) = \begin{cases}0 & if q = 0, \\ |Γ_{1}^{0}G|/|γ_{2}^{q}(M^{0}(G))| & if q ≥ p, \end{cases}

where H_{n}(G, ℤ_{q}) is the n-th homology group of G with coefficients in the abelian group ℤ_{q} = ℤ/qℤ. Thus Z^{0}G is the centre of G, γ_{1}^{0}G ≤ γ_{2}^{0}G ≤ ⋯ is the lower central series of G, and when G is finite M^{0}(G) is the Schur multiplier of G. If G is a finite p-group, then γ_{2}^{p}G is its Frattini subgroup, and Γ_{i}^{p}G is an elementary abelian group for i ≥ 1. We say that a group is a “d-generator group” if it can be generated by d elements, and by no fewer than d elements.

Theorem 1. Let G be any group such that Q = G/Z^{q}(G) is a d-generator group of order p^{n}. Then:

(i) |γ_{2}^{q}G| ≤ p^{(d(2n−d−1)/2)+R(Q,q)}. (ii) There exists a group Ĝ with Ĝ/Z^{q}(Ĝ) ≅ Q and |γ_{2}^{q}Ĝ| = p^{(d(2n−d−1)/2)+R(Q,q)}

if and only if the canonical homomorphism H_{3}^{0}(Q) → H_{3}^{0}(Q/γ_{i}^{p}Q) is surjective for each i ≥ 1.

(iii) If d = n ≥ 2, or if d = n − 1 = 2, then there exists a group Ĝ with Ĝ/Z^{q}(Ĝ) a d-generator group of order p^{n}, and such that |γ_{2}^{q}Ĝ| = p^{(d(2n−d−1)/2)+R(Q,q)}.

Note that p^{d(2n−d−1)/2} ≤ p^{n(n−1)/2} for all integers 1 ≤ d ≤ n. Thus Wiegold’s bound [17, Theorem 2.1] on the commutator subgroup γ_{2}^{0}G follows from Theorem 1.

Note that part (iii) of Theorem 1 does not extend to d = n − 2 = 2. Consider, for instance, the values p = 2, q = 0, n = 4, d = 2 and consider any group G with G/Z^{0}(G) a 2-generator group of order p^{4}. Arguments involving the Schur multiplier (see [2]) establish the existence of such a G, and show that the only possibilities for the commutator subgroup are γ_{2}^{0}G ≅ C_{2} × C_{2}, γ_{2}^{0}G ≅ C_{2} × C_{4} and γ_{2}^{0}G ≅ C_{2} × C_{2} × C_{4}; in all cases |γ_{2}^{0}G| is strictly less than p^{d(2n−d−1)/2} = 32.

The next result is a variant on Theorem 1 which incorporates the structure of the abelian groups (Q/Z^{0}(Q))^{ab} and Γ_{i}^{0}(Q) for i ≥ 1.

Theorem 2. Let G be any group with |G/Z^{q}(G)| = p^{n}. Set Q = G/Z^{q}(G) and \overline{Q} = Q/Z^{0}(Q), and suppose that

Γ_{1}^{0}\overline{Q} ≅ C_{p^{m{′ \atop 11}}} × C_{p^{m{′ \atop 12}}} × ⋯ × C_{p^{m{′ \atop 1r}}} ,
Γ_{i}^{0}Q ≅ C_{p^{m_{i1}}} × C_{p^{m_{i2}}} × ⋯ × C_{p^{m_{ir}}}

for i ≥ 1. (Here r is independent of i and m_{ij} ≥ 0.) Set

M = ∑_{1≤j<k≤r} minimum(m_{1j}, m_{1k}) + ∑_{\begin{matrix}2≤i≤r \\ 1≤j, k≤r \end{matrix}} minimum(m_{ij}, m′_{1k}).

Then (i) |γ_{2}^{q}G| ≤ p^{M+R(Q,q)}. (ii) There exists a group Ĝ with Ĝ/Z^{q}(Ĝ) ≅ Q and |γ_{2}^{q}Ĝ| = p^{M+R(Q,q)} if the

canonical homomorphism H_{3}^{0}(Q) → H_{3}^{0}(Q/γ_{i}^{0}Q) is surjective for each i ≥ 1.

This theorem improves slightly on a bound of Gaschütz, Neubüser and Yen [6]. They proved that if Q = G/Z^{0}(G) is a prime-power group whose central quotient Q/Z^{0}(Q) is generated by d elements, then

|[G, G]| ≤ |M^{0}(Q^{ab})| |[Q, Q]|^{d}.

To see the improvement, note that M^{0}(Q^{ab}) is just the exterior square Q^{ab} ∧ Q^{ab} and hence that

log_{p} |M^{0}(Q^{ab})| = ∑_{1≤j<k≤r} minimum(m_{1j}, m_{1k}).

Furthermore, if Q is nilpotent of class c say, then equality (4) below (which involves the tensor product of abelian groups) yields

log_{p} |[Q, Q]|^{d} = log_{p} |Γ_{2}^{0}Q ⊕ ⋯ ⊕ Γ_{c}^{0}Q|^{d}

≥ log_{p} |(Γ_{2}^{0}Q ⊕ ⋯ ⊕ Γ_{c}^{0}Q) ⊗ Γ_{1}^{0}\overline{Q}|

= ∑_{\begin{matrix}2≤i≤r \\ 1≤j, k≤r \end{matrix}} minimum(m_{ij}, m′_{1k}).

The last inequality is strict precisely when the exponent of Γ_{2}^{0}Q ⊕ ⋯ ⊕ Γ_{c}^{0}Q is greater than the order of at least one of the d generators of Γ_{1}^{0}\overline{Q} = Q/[Q, Q]Z^{0}(Q); in this case the bound in Theorem 2 is lower than the bound of [6].

To illustrate the above results set p = 2, q = 0, and consider the semi-direct product Q = ⟨x, y|x^{8} = y^{4} = 1, xy = yx^{3}⟩ of C_{4} with C_{8}, which has order 2^{5}. Let G be any group with Q ≅ G/Z^{0}(G). Arguments involving the Schur multiplier of Q (cf. [2]) show that, in this case, such a group G exists and that γ_{2}^{0}G must have order 2^{3}. Wiegold’s bound asserts that

|γ_{2}^{0}G| ≤ 2^{n(n−1)/2} = 2^{10}.

The bound of Gaschütz, Neubüser and Yen asserts that

|γ_{2}^{0}G| ≤ |M^{0}(C_{2} × C_{4})| |C_{2} × C_{2}|^{2} = 2^{5}.

Since Q is a 2-generator group, Theorem 1(i) asserts that

|γ_{2}^{0}G| ≤ 2^{d(2n−d−1)/2} = 2^{7}.

Since γ_{2}^{0}Q/γ_{3}^{0}Q ≅ C_{2}, γ_{3}^{0}Q ≅ C_{2}, and γ_{i}^{0}Q = 1 for i ≥ 4, and since (Q/Z^{0}(Q))^{ab} ≅ C_{4} × C_{2}, Theorem 2(i) asserts that

|γ_{2}^{0}G| ≤ 2^{M} = 2^{5}.

Let Q be any finite p-group. Then γ_{c+1}^{p}Q = 1 for some c ≥ 1, and the direct sum 𝔏 = Γ_{1}^{p}Q ⊕ Γ_{2}^{p}Q ⊕ ⋯ ⊕ Γ_{c}^{p}Q has the structure of a restricted Lie algebra over the field 𝔽 = ℤ_{p} (cf. [12]). The Lie bracket 𝔥: 𝔏 × 𝔏 → 𝔏 is induced by commutation in Q, and the power map 𝔭: 𝔏 → 𝔏 is induced by taking p-th powers in Q. For each i ≥ 1 the bracket 𝔥 restricts to a linear homomorphism 𝔥: Γ_{i}^{p}Q ⊗ Γ_{1}^{p}Q → Γ_{i+1}^{p}Q of vector spaces, and repetition of the power map 𝔭 yields a linear homomorphism 𝔭^{i}: Γ_{1}^{p}Q → Γ_{i+1}^{p}Q. The bound of Theorem 1 can be improved by considering the following linear homomorphisms:

𝔧_{2} : Γ_{1}^{p}Q ⊗ Γ_{1}^{p}Q ⊗ Γ_{1}^{p}Q → Γ_{2}^{p}Q ⊗ Γ_{1}^{p}Q,

x ⊗ y ⊗ z ↦ 𝔥(x, y) ⊗ z + 𝔥(y, z) ⊗ x + 𝔥(z, x) ⊗ y,

𝔧_{i}: Γ_{1}^{p}Q ⊗ Γ_{1}^{p}Q ⊗ ⋯ ⊗ Γ_{1}^{p}Q → Γ_{i}^{p}Q ⊗ Γ_{1}^{p}Q,

x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{i+1} ↦ [x_{1}, ⋯, x_{i}]_{l} ⊗ x_{i+1} + [x_{i+1}, [x_{1}, ⋯, x_{i−1}]_{l}]
⊗ x_{i} + [[x_{i}, x_{i+1}]_{r}, [x_{1}, ⋯, x_{i−2}]_{l}]
⊗ x_{i−1} + [[x_{i−1}, x_{i}, x_{i+1}]_{r}, [x_{1}, ⋯, x_{i−3}]_{l}]
⊗ x_{i−2} + ⋯ + [x_{2}, ⋯, x_{i+1}]_{r} ⊗ x_{1},

𝔰_{i}: Γ_{1}^{p}Q → Γ_{i}^{p}Q ⊗ Γ_{1}^{p}Q, x ↦ (𝔭^{i−1}x) ⊗ x, i ≥ 2,

𝔱_{i}: Γ_{1}^{p}Q ⊗ Γ_{1}^{p}Q → Γ_{i}^{p}Q ⊗ Γ_{1}^{p}Q, x ⊗ y ↦ (𝔭^{i−1}x) ⊗ y + (𝔭^{i−1}y) ⊗ x, i ≥ 2.

Here x_{i} ∈ Γ_{1}^{p}Q and we have set

[x_{1}, x_{2}] = 𝔥(x_{1} ⊗ x_{2}), [x_{1}, ⋯, x_{i}]_{l}
= 𝔥(⋯ 𝔥(𝔥(x_{1} ⊗ x_{2}) ⊗ x_{3}) ⋯ ⊗ x_{i}), [x_{1}, ⋯, x_{i}]_{r}
= 𝔥(x_{1} ⊗ ⋯ 𝔥(x_{i−2} ⊗ 𝔥(x_{i−1} ⊗ x_{i})) ⋯).

Note that 𝔧_{2}(x, y, z) = 0 whenever two of x, y, z are equal. Hence 𝔧_{2} has trivial image whenever Q is a 2-generator group.

Theorem 3. Let G be any group such that Q = G/Z^{q}(G) is a d-generator group of order p^{n}. Set

a_{i} = dim_{𝔽}(image(𝔧_{i}) + image(𝔰_{i}) + image(𝔱_{i}))

for i ≥ 2, and set a = a_{1} + a_{2} + ⋯ + a_{c} where γ_{c+1}^{p}Q = 1. Then

|γ_{2}^{q}G| ≤ p^{(d(2n−d−1)/2)−a+R(Q,q)}.

To illustrate Theorem 3 let us return to a group G such that G/Z^{0}(G) ≅ Q = ⟨x, y|x^{8} = y^{4} = 1, xy = yx^{3}⟩. Now Γ_{1}^{2}Q ≅ ℤ_{2} ⊕ ℤ_{2} is generated by the cosets of x and y; Γ_{2}^{2}Q ≅ ℤ_{2} ⊕ ℤ_{2} is generated by the cosets of x^{2} and y^{2}; Γ_{3}^{2}Q ≅ Z_{2} is generated by the coset of x^{4}; Γ_{i}^{2}Q = 0 for i ≥ 4. Since Q is a 2-generator group, 𝔧_{2} has trivial image. One readily verifies that 𝔧_{3} also has trivial image. Thus a_{2} = dim(image(𝔰_{2}) + image(𝔱_{2})) = 3, a_{3} = dim(image(𝔰_{3}) + image(𝔱_{3})) = 1, and a = 3 + 1 = 4. Therefore Theorem 3 asserts that

|γ_{2}^{0}G| ≤ 2^{(d(2n−d−1)/2)−a} = 2^{3}.

Vaughan-Lee [16] showed that the commutator subgroup of a finite p-group of “breadth” b has order at most p^{b(b+1)/2}. Recall that the breadth b(x) of an element x in an arbitrary group G is defined (with respect to the prime p) as the logarithm (to the base p) of the number of elements conjugate to x:

b(x) = log_{p} |[x]|.

Of course, b(x) is infinite when the conjugacy class [x] is infinite. The breadth of the group G is defined as the maximum value of b(x) for x ∈ G if such a maximum exists; otherwise it is defined to be infinite.

Suppose that Q = G/Z^{0}(G) is a finite p-group. If C(x) denotes the centralizer of x in G, then p^{b(x)} = |G: C(x)|. Since |Q| = |G: Z^{0}(G)| = |G: C(x)| × |C(x) :

Z^{0}(G)| is a power of p, it follows that b(x) is a nonnegative integer for all x in G. For any finite subset X = {x_{1}, …, x_{d}} in G, set

b(X) = (b(x_{1}) + ⋯ + b(x_{d}))/d,

and let \overline{X} denote the image of X in Q. It seems reasonable to define the average breadth of G to be

\overline{b}(G) = minimum\{b(X): \overline{X} generates G/Z^{0}(G)\}.

The following theorem complements Vaughan-Lee’s result.

Theorem 4. Let G be any group such that Q = G/Z^{0}(G) is a d-generator finite p-group. If G has average breadth \overline{b} = \overline{b}(G), then

|γ_{2}^{0}G| ≤ p^{d\overline{b}}.

Our proofs of the above theorems yield the following result of independent interest.

Proposition 5. Let Q be any d-generator group of order p^{n}.

(i) For a defined as in Theorem 3, we have

|M^{q}(Q)| |γ_{2}^{q}Q| ≤ p^{(d(2n−d−1)/2)−a+R(Q,q)}.

(ii) If the abelian groups (Q/Z^{0}(Q))^{ab} and Γ_{i}^{0}Q have structure as in Theorem 2, then

|M^{q}(Q)| |γ_{2}^{q}Q| ≤ p^{M+R(Q,q)}.

This bound is attained if the canonical homomorphism H_{3}^{0}(Q) → H_{3}^{0}(Q/γ_{i}^{0}Q) is surjective for each i ≥ 1.

Proposition 5(i) (with q = 0) sharpens a bound on the Schur multiplier given in [9, Corollary 2.3]. The bound in Proposition 5(ii) (with q = 0) is similar to, but lower than, a bound given in [10, Theorem 4.1] (see also [11]); the latter bound uses γ_{i}^{0}G/[γ_{i}^{0}G, γ_{i}^{0}G] in place of our γ_{i}^{0}G/γ_{i+1}^{0}G. Proposition 5(ii) also sharpens a bound given in [6].

The proof of Proposition 5 yields analogous bounds on the exponent of M^{q}(Q) and on the number of generators of M^{q}(Q). Details of these are left to the reader.

Proposition 5 is relevant to the enumeration of finite p-groups. Let f(n, p) denote the number of groups of order p^{n}, and set A(n, p) = log_{p}(f(n, p)). Higman [8] proved that

(2/27 − ε_{n})n^{3} ≤ A(n, p) ≤ (n^{3} − n)/6

where ε_{n} → 0 as n → ∞. Sims [14] subsequently showed that A(n, p)/n^{3} → 2/27 as n → ∞. In order to get a feel for the types of p-groups that occur one should investigate, for various group-theoretic properties P, the number f(n, p, P) of groups of order p^{n} satisfying property P.

Let us consider the number f(n, p, d) of d-generator groups of order p^{n}, and set

A(n, p, d) = log_{p}(f(n, p, d)).

A lower bound for A(n, p, d) is implicit in [8]. An upper bound can be obtained from Proposition 5. To this effect let us define

U(n, d) = \begin{cases}(d − 1)(n^{2} − d^{2})/2, & for d ≠ 2, \\ n(n + 1) − 6, & for d = 2, \end{cases}

L(n, d) = (d(d + 1)(n − d) − 2(n − d)^{2} − 2d^{2})/2.

The following corollary shows that these two functions give reasonable bounds on A(n, p, d) for “large” values of d.

Corollary 6. For all integers n ≥ d ≥ 1 we have

L(n, d) ≤ A(n, p, d) ≤ U(n, d).

Furthermore, A(d, p, d) = U(d, d) = 0 and for any integer t ≥ 2 we have

lim_{d→∞} \frac{U(td, d)}{L(td, d)} = t + 1.

Proof. It is implicit in [8] (see also [15]) that L(n, d) is less than A(n, p, d). So we shall concentrate on the upper bound, and on the case d ≠ 2 (leaving to the reader the case d = 2). Certainly A(n, p, 1) = U(n, 1), so let us suppose that d ≥ 3. Clearly A(d, p, d) = U(d, d). As an inductive hypothesis assume that A(k, p, d) ≤ U(k, d) for some k ≥ d. For each group G of order p^{k+1} we can choose a central subgroup Z of order p. The group G is determined, up to an extension, by Z and G/Z. In other words G is determined by Z, G/Z, and an element of the second cohomology group H^{2}(G/Z, ℤ_{p}). The Universal Coefficient Theorem

[figure: Ext((G/Z)^{ab}, ℤ_{p}); H^{2}(G/Z, ℤ_{p}); Hom(M^{0}(G/Z), ℤ_{p})]

and Proposition 5(i) imply that

|H^{2}(G/Z, ℤ_{p})| ≤ |M^{0}(G/Z)| · p^{d} ≤ p^{(d(2k−d+3)−2k)/2}.

Therefore

A(k + 1, p, d) ≤ U(k, d) + log_{p} |H^{2}(G/Z, ℤ_{p})|
≤ ((d − 1)(k^{2} − d^{2}) + (d(2k − d + 3) − 2k))/2
≤ ((d − 1)((k + 1)^{2} − d^{2}) − d^{2} + 2d + 1)/2
≤ ((d − 1)((k + 1)^{2} − d^{2}))/2
≤ U(k + 1, d).

The upper bound of the proposition follows by induction. It is routine to verify the limit in the proposition.

Our proofs of the above theorems involve the q-exterior product M∧^{q}N of two normal subgroups M and N of some group, a self-contained account of which can be found in [4]. We shall assume a familiarity with the definition of this q-exterior product, and with certain results from [4]. For q = 0 the exterior product M∧^{0}N is that of Brown and Loday [1].

It is shown in [4, p. 247] that M∧^{q}N is finite if both M and N are finite. The following lemma relates the cardinality of the exterior square Q∧^{q}Q to the above theorems.

Lemma 7. Let G be any group and set Q = G/Z^{q}(G). Then:

(i) |γ_{2}^{q}G| ≤ |Q∧^{q}Q|.

Moreover, there is a group Ĝ such that Ĝ/Z^{q}(Ĝ) ≅ Q and γ_{2}^{q}Ĝ ≅ Q ∧^{q} Q. (ii) |M^{q}(G)| |γ_{2}^{q}G| = |G∧^{q}G|.

Proof. Theorem 6 and Proposition 7 in [4] (the first of these results being due to [3]) yield the following diagram of solid homomorphisms:

\xymatrix{Z^{q}(G)∧^{q}G \ar[r]^{ι} & G∧^{q}G \ar[d]_{∂} \ar[r]^{π} & Q∧^{q}Q \ar@{.>}[dl] \ar[r] & 1 \\  & G \ar[d] \\  & Γ_{1}^{q}G \ar[d] \\  & 1}

in which the row and column are exact, and in which im(ι) ⊆ ker(∂). There is thus a surjection Q∧^{q}Q ↠ γ_{2}^{q}G from which we deduce that |γ_{2}^{q}G| ≤ |Q∧^{q}Q|.

Let μ: Ĝ → Q be any projective q-crossed module with im(μ) = Q; the definition and existence of such an object is given in [4, p. 245]. Proposition 5 in [4] implies that γ_{2}^{q}Ĝ ≅ Q∧^{q}Q. The proof of Proposition 16(vii) in [4] implies that Ĝ/Z^{q}(Ĝ) ≅ Q.

Propositions 1 and 5 in [4] imply an exact sequence (which is also to be found in [5])

1 → M^{q}(G) → G∧^{q}G → G → Γ_{1}^{q}G → 1.

The equality |M^{q}(G)| |γ_{2}^{q}G| = |G∧^{q}G| follows from the exactness of this sequence.

In the following lemma (and throughout the article) we use the symbol ⊗ to denote the usual tensor product of abelian groups considered as ℤ-modules.

Lemma 8. (i) [1] Any normal inclusion N ≤ G gives rise to a natural exact sequence

H_{3}^{0}(G) → H_{3}^{0}(G/N) → N∧^{0}G → G∧^{0}G → (G/N)∧^{0}(G/N) → 1.

(ii) [4] Any pair of normal inclusions N ≤ G and M ≤ G with M ⊆ N give rise to a natural exact sequence

M∧^{0}G → N∧^{0}G → (N/M)∧^{0}(G/M) → 1.

(iii) There is an isomorphism

(Γ_{i}^{p}G)∧^{0}(G/γ_{i+1}^{p}G) ≅ (Γ_{i}^{p}G) ⊗ (Γ_{1}^{p}G)

for i ≥ 2. (iv) For q ≥ p there is an exact sequence

0 → γ_{2}^{q}(M^{0}(G)) → G∧^{0}G → G∧^{q}G → Γ_{1}^{0}G → 0.

Proof. Part (i) can be found, for instance, in [1], and part (ii) is an easy modification of Proposition 7 in [4]. To prove (iii) note that Γ_{i}^{p}G lies in the centre of G/γ_{i+1}^{p}G. As explained in [1], in this case the defining relations of the exterior product yield an isomorphism

Γ_{i}^{p}G∧^{0}(G/γ_{i+1}^{p}G) ≅ Γ_{i}^{p}G ⊗ (G/γ_{i+1}^{p}G)^{ab}/⟨x ⊗ \overline{x} | x ∈ Γ_{i}^{p}G⟩

where \overline{x} denotes the image of x in the abelianization of G/γ_{i+1}^{p}G. The isomorphism of (iii) follows. The exact sequence of (iv) follows from Proposition 14 in [4].

Set ε = 0 or ε = p. Let Q be a p-group of order p^{n}. Let c be the first integer such that then γ_{c+1}^{ε}Q = 1.

Lemma 8(i) applied to the normal inclusion γ_{2}^{ε}Q ≤ Q yields the exact sequence

\xymatrix{γ_{2}^{ε}Q∧^{0}Q \ar[r]^{α_{2}} & Q∧^{0}Q \ar[r] & Γ_{1}^{ε}Q∧^{0}Γ_{1}^{ε}Q \ar[r] & 1.} (1)

For i ≥ 2 the normal inclusion γ_{i+1}^{ε}Q ≤ γ_{i}^{ε}Q in conjunction with Lemma 8(ii) yields the exact sequence

\xymatrix{γ_{i+1}^{ε}Q∧^{0}Q \ar[r]^{α_{i+1}} & γ_{i}^{ε}Q∧^{0}Q \ar[r] & (γ_{i}^{ε}Q/γ_{i+1}^{ε}Q)∧^{0}(Q/γ_{i+1}^{ε}Q) \ar[r] & 1.} (2)

Sequences (1) and (2) in conjunction with Lemma 8(iii) imply

(3) |Q∧^{0}Q| = \frac{|Γ_{1}^{ε}Q∧^{0}Γ_{1}^{ε}Q| |Γ_{2}^{ε}Q ⊗ Γ_{1}^{ε}Q| ⋯ |Γ_{c−1}^{ε}Q ⊗ Γ_{1}^{ε}Q| |γ_{c}^{ε}Q ⊗ Γ_{1}^{ε}Q|}{| ker α_{2}| | ker α_{3}| ⋯ | ker α_{c}|}.

Recall that the tensor product of any two abelian groups can be computed from the following isomorphisms: A ⊗ (B × C) ≅ (A ⊗ B) × (A ⊗ C), A ⊗ B ≅ B ⊗ A, and C_{m} ⊗ C_{n} ≅ C_{hcf(m,n)} where A, B, C are arbitrary abelian groups, and C_{m} is the cyclic group of order m. Recall from [1] that for any abelian group A there is an isomorphism A∧^{0}A ≅ A ⊗ A/⟨a ⊗ a : a ∈ A⟩. Thus, if

Γ_{i}^{ε}Q ≅ C_{p^{m_{i1}}} × C_{p^{m_{i2}}} × ⋯ × C_{p^{m_{ir}}}

and if M_{ijk} = minimum(m_{ij}, m_{1k}), then

log_{p} |Γ_{1}^{ε}Q∧^{0}Γ_{1}^{ε}Q| = ∑_{1≤j<k≤r} M_{1jk},

log_{p} |Γ_{i}^{ε}Q ⊗ Γ_{1}^{ε}Q| = ∑_{1≤j, k≤r} M_{ijk}, i ≥ 2.

(4)

Note that each |Γ_{i}^{ε}Q ⊗ Γ_{1}^{ε}Q| is a maximum, for any given |Γ_{i}^{ε}Q| = p^{m_{i}} and |Γ_{1}^{ε}Q| = p^{m_{1}}, if and only if Γ_{i}^{ε}Q and Γ_{1}^{ε}Q are elementary abelian; in other words

\begin{matrix}|Γ_{1}^{ε}Q∧^{0}Γ_{1}^{ε}Q| ≤ p^{m_{1}(m_{1}−1)/2}, \\ |Γ_{i}^{ε}Q ⊗ Γ_{1}^{ε}Q| ≤ p^{m_{1}m_{i}}, i ≥ 2. \end{matrix} (5)

Now (3) and (5) imply that

\begin{aligned}|Q∧^{0}Q| & ≤ p^{m_{1}(m_{1}−1)/2+m_{1}m_{2}+m_{1}m_{3}+···m_{1}m_{c}} \\ & = p^{m_{1}(m_{1}−1+2m_{2}+2m_{3}+···+2m_{c})/2} \\ & = p^{m_{1}(2n−m_{1}−1)/2}. \end{aligned} (6)

Note that if ε = p, then m_{1} = d is the minimum number of generators needed for Q. Thus Theorem 1(i) follows from (6), the inequality of Lemma 7(i), and the exact sequence of Lemma 8(iv). For i ≥ 2 Lemma 8(i) yields an exact sequence

\begin{aligned}H_{3}^{0}(Q) & ⟶ H_{3}^{0}(Q/γ_{i}^{ε}Q) ⟶ γ_{i}^{ε}Q∧^{0}Q ⟶^{β_{i}} Q∧^{0}Q \\ & ⟶ (Q/γ_{i}^{ε}Q) ∧^{0} (Q/γ_{i}^{ε}Q) ⟶ 1. \end{aligned}

Now β_{i} factors as

\xymatrix{β_{i}: γ_{i}^{ε}Q∧^{0}Q \ar[r]^{α_{i}} & γ_{i−1}^{ε}Q∧^{0}Q \ar[r] & ⋯ \ar[r]^{α_{2}} & Q∧^{0}Q.}

So ker β_{i} = 0 for all i ≥ 2 if and only if ker α_{i} = 0 for all i ≥ 2. This equivalence, together with Lemma 7(i) and our derivation of Theorem 1(i), imply Theorem 1(ii).

In order to prove Theorem 2 note that when ε = 0 and i ≥ 2 the homomorphism β_{i} factors as

\xymatrix{γ_{i}^{0}Q∧^{0}Q \ar[dr] \ar[rr]^{β_{i}} &  & Q∧^{0}Q \\  & γ_{i}^{0}Q∧^{0}(Q/Z^{0}Q) \ar[ur]}

since the kernel of the canonical homomorphism γ_{i}^{0}Q∧^{0}Q → γ_{i}^{0}Q∧^{0}(Q/Z^{0}Q) is generated by the elements x∧z with x ∈ γ_{i}^{0}Q and z ∈ Z^{0}(Q), and since β_{i}(x∧z) = 1 for each such generator. (To prove the triviality of β_{i}(x ∧ z), note that we can find an element x̃ ∈ Q∧^{0}Q which maps canonically onto x. We have β_{i}(x ∧ z) = x ∧ z = x̃^{z}x̃^{−1} by [1]. But z is central and thus acts trivially on Q∧^{0}Q. Therefore β_{i}(x ∧ z) = x̃^{z}x̃^{−1} = 1.) This factorization, together with the arguments behind (3), yield the inequality

|Q∧^{0}Q| ≤ |Γ_{1}^{0}Q∧^{0}Γ_{1}^{0}Q| |Γ_{2}^{0}Q ⊗ Γ_{1}^{0}\overline{Q}| ⋯ |Γ_{c−1}^{0}Q ⊗ Γ_{1}^{0}\overline{Q}| |γ_{c}^{ε}Q ⊗ Γ_{1}^{0}\overline{Q}| (7)

where \overline{Q} = Q/Z^{0}(Q).

The proof of Theorem 2 is similar to that of Theorem 1; the differences are that now ε = 0, and that (7) can be used in place of (3).

Theorem 3 is an improvement on parts (i) and (ii) of Theorem 1 obtained by using Theorem 9 in [4] to identify certain non-trivial elements in ker α_{i}, and thus to estimate | ker α_{i}| in (3).

Proposition 5 follows in the same fashion from (3), (7) and Lemma 7(ii).

To prove Theorem 4 suppose that Q = G/Z^{0}(G) is of order p^{n}, and that X = {x_{1}, …, x_{d}} is a subset of G whose image in Q generates Q, and which satisfies \overline{b}(G) = b(X). Denote by C_{i} the image in Q of the centraliser C(x_{i}) of x_{i} in G. Denote by U_{i} the cyclic subgroup of Q generated by the image of x_{i}. The image of the canonical homomorphism C_{i}∧^{0}U_{i} → Q∧^{0}Q clearly lies in the kernel of the surjection Q∧^{0}Q ↠ γ_{2}^{0}G, and this observation is the basis of the proof. Set ε = p, let V_{i} denote the cyclic summand of Q/γ_{2}^{ε}(Q) generated by the image of x_{i}, and set D_{ij} = C_{i} ∩ γ_{j}^{ε}(Q). Note that the canonical homomorphisms

[figure: (D_{ij}/D_{ij+1}) ⊗ V_{i}; Γ_{j}^{ε}Q ⊗ Γ_{1}^{ε}Q]

are injective, and that

|(D_{i1}/D_{i2}) ⊗ V_{i}| |(D_{i2}/D_{i3}) ⊗ V_{i}| ⋯ |D_{ic} ⊗ V_{i}| = |C_{i}| = p^{n−b(x_{i})}.

Combining these three observations with (3) yields

|γ_{2}^{0}(G)| ≤ \frac{|Γ_{1}^{ε}Q ⊗ Γ_{1}^{ε}Q| |Γ_{2}^{ε}Q ⊗ Γ_{1}^{ε}Q| ⋯ |Γ_{c−1}^{ε}Q ⊗ Γ_{1}^{ε}Q| |γ_{c}^{ε}Q ⊗ Γ_{1}^{ε}Q|}{∏_{1≤i≤d} |(D_{i1}/D_{i2}) ⊗ V_{i}| |(D_{i2}/D_{i3}) ⊗ V_{i}| ⋯ |(D_{ic} ⊗ V_{i})|}
≤ p^{nd}/p^{n−b(x_{1})+···+n−b(x_{d})}
= p^{d\overline{b}(G)}

as required.

To prove Theorem 1(iii) for d = n ≥ 2 set Q equal to the elementary abelian p-group on n generators. Then |Q| = |Γ_{1}^{q}Q| = p^{n}, R(Q, q) = 0, and |Q∧^{q}Q| = p^{(n(n−1)/2)}. Moreover, Q ≅ G/Z^{q}(G) for some G by Proposition 16(vii) in [4]. Thus the second assertion of Lemma 7(i) establishes the existence of a group Ĝ with Ĝ/Z^{q}(Ĝ) ≅ Q and |γ_{2}^{q}Ĝ| = p^{(n(n−1)/2)+R(Q,q)}.

To prove Theorem 1(iii) for d = n − 1 = 2 let us first note that for any abelian groups A, B we can construct a semi-direct product

A□B = ((A ⊗ B) × B) ⋊ Z

in which the action of an element a in A on the action of an element (τ, b) in the direct product (A ⊗ B) × B is given by

a(τ, b) = (τ(a ⊗ b), b).

If the elements of A and B all have order dividing q, then

γ_{2}^{q}(A□B) ≅ A ⊗ B.

To prove Theorem 1(iii) for d = n − 1 = 2 set Q = C_{p}□C_{p}. Then |Q| = p^{3}, |Γ_{1}^{q}Q| = p^{2}, and one can readily verify that |Q∧^{q}Q| = p^{3+R(Q,q)} = p^{(d(2n−d−1)/2)+R(Q,q)}. Now Q ≅ G/Z^{q}(G) for some G by Proposition 16(vii) in [4]. Thus the second assertion of Lemma 7(i) establishes the existence of a group Ĝ with Ĝ/Z^{q}(Ĝ) ≅ Q and |γ_{2}^{q}Ĝ| = p^{(d(2n−d−1)/2)+R(Q,q)}.

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Department of Mathematics, University College, Galway, Ireland

   E-mail address: graham.ellis@ucg.ie

Current address, September 1998 to June 1999: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, Bonn, Germany