A connection between nilpotent groups and Lie rings

A. Jaikin-Zapirain^{1} and E. I. Khukhro^{2}

Introduction

Let G be a nilpotent group of class c. We use the Baker–Hausdorff formula to define the structure of a Lie ring (ℤ-algebra) M on the subgroup G^{n}, for some n = n(c) depending only on c, in such a way that many important parameters of M, like the nilpotency class and the derived length, are equal to those of G^{n} as a group.

As an application we refine reductions of theorems about “almost regular” p-automor-phisms of finite p-groups to corresponding theorems on Lie rings. In particular, we prove that the m-bounded function in Medvedev’s theorem [15] on p-groups can be chosen to be exactly the same as in his theorem on Lie rings. We also give a new version of a part of the proof of Khukhro’s theorem [7] (this proof was one of the origins of the present paper). Besides, we show that Higman’s and Kreknin’s functions that appear in results on fixed-point-free automorphisms of Lie algebras are the best possible bounds (if required to depend only on the order of the automorphism) for the nilpotency class and the derived length respectively of a subgroup of bounded index in theorems on p-automorphisms of finite p-groups. (We do not obtain any better estimates for Higman’s or Kreknin’s functions for Lie algebras; our purpose is to improve, up to best-possible, the bounds for groups in terms of these functions.)

The use of the exponential mapping and the Baker–Hausdorff formula H(x, y) = log(e^{x}e^{y}) goes back to the theory of Lie groups. For discrete groups the Baker–Hausdorff formula is used in the Mal’cev correspondence, which is an equivalence of categories of (locally) nilpotent ℚ-groups (that is, torsion-free and divisible) and (locally) nilpotent Lie ℚ-algebras. Namely, for any nilpotent Lie ℚ-algebra M the same set becomes a nilpotent ℚ-group G = M with respect to operations a · b := H(a, b) and a^{r} := ra, r ∈ ℚ, with operations in M on the right. The inversions of the Baker–Hausdorff formula express the Lie ℚ-algebra operations in M via the ℚ-group operations in G. This correspondence enables one to prove results about nilpotent groups using Lie algebras; of course, the converse can also be said, but Lie algebras often seem easier to deal with, because of the rich linear structure. For example, one can thus prove a folklore theorem that a torsion-free (locally) nilpotent group with a fixed-point-free automorphism of finite order n is soluble of derived length bounded in terms of n only. Namely, by the Mal’cev correspondence one passes to Lie algebras, for which the corresponding result is due to Kreknin [9] (whose proof relies on extending the ground field and calculations with the eigen-spaces which form a (ℤ/nℤ)-grading).

However, this technique can not be directly applied to finite p-groups, which are neither torsion-free nor divisible. Every nilpotent group has the so-called associated Lie ring, but it does not adequately reflect some important features of the group, for example, its derived length. Lazard’s correspondence [11] for p-groups, also based on the Baker– Hausdorff formula, requires quite strong restrictions on the p-group to be applicable, for example, the nilpotency class to be less than p. Recently Weigel [17] applied the Lazard correspondence (in its more general setting) to p-groups for p ≥ 5 that are both powerful and p-central (this class includes uniformly powerful p-groups).

1The work of A. Jaikin-Zapirain has been supported by a grant of the Government of the Basque Country and by the DGICYT grant PB97–0604

2The work of E. I. Khukhro has been supported by the Russian Foundation for Basic Research, grant 99-01-00576.

Our correspondence can be applied to any nilpotent group, including finite p-groups, of any nilpotency class c, but it is based on a subgroup of the form G^{n}, for some n = n(c) depending on c. It may not be as good as the Mal’cev correspondence; for example, not every subgroup of G^{n} corresponds to a Lie subring, etc. But it does preserve many important parameters of G^{n} and the corresponding Lie ring, like the derived length and the nilpotency class. In applications to p-automorphisms of finite p-groups the (sectional) rank of the group is often a bounded parameter, so that the subgroup G^{n(c)} has then bounded index, provided, of course, that we already succeeded in bounding the nilpotency class.

1. Preliminary lemmas

To distinguish group and Lie ring commutators we shall use lower indices _{G} and _{L} respectively. For example, by the Baker–Hausdorff formula [x, y]_{G} = [x, y]_{L} + …, where the dots denote rational linear combinations of Lie ring commutators in the variables x and y. Similarly, we shall use the signs ⊲_{G} for normal subgroups and ⊲_{L} for ideals, and ⟨S⟩_{G} or ⟨S⟩_{L} for a subgroup or a Lie subring generated by the set S. We use standard notation _{id}⟨S⟩ for the ideal generated by S and ⟨S⟩^{T} for the normal closure of S in T.

Let L be the free nilpotent Lie ring (ℤ-algebra) of class c on the set of free generators x_{1}, x_{2}, … (not necessarily countable) naturally embedded into the free nilpotent Lie ℚalgebra ℚL on the same set of free generators. The Baker–Hausdorff formula H(x, y) defines the structure of a nilpotent ℚ-group F̂ on the same set ℚL, with respect to operations a · b := H(a, b) and a^{r} := ra, r ∈ ℚ, with operations in ℚL on the right (see, for example, [3]). As a ℚ-group F̂ is free nilpotent of class c on the free generators x_{1}, x_{2}, …. The inversions of the Baker–Hausdorff formula express the Lie ℚ-algebra operations in ℚL via the ℚ-group operations in F̂. Let F be the abstract subgroup of F̂ generated by x_{1}, x_{2}, …; then F is a free nilpotent group on free generators x_{1}, x_{2}, …. Every element of F is some power of an element of F̂ for some k. Every element g ∈ F̂ has a unique presentation g = ∏ c_{i}^{r_{i}}, where r_{i} ∈ ℚ and the c_{i} are basic group commutators in the x_{i} in some fixed order, and g ∈ F if and only if r_{i} ∈ ℤ for all i. Without insisting on uniqueness, one can rewrite any element of F̂ as a product of rational powers of simple commutators in the x_{i}.

In the proofs we shall often use the following argument based on the fact that F, F̂, L and ℚL are free objects in respective categories. Suppose that we have established the equality v(x_{1}, …, x_{k}) = w(x_{1}, …, x_{k}) for some words v and w in free generators x_{i}. Then v(a_{1}, …, a_{k}) = w(a_{1}, …, a_{k}) for any elements a_{i} of any object in the same category, by virtue of the homomorphism extending the map x_{i} → a_{i}. If we stay within ℚL = F̂, then v and w can even be words in different categories: since the operations in F̂ are expressed as fixed formulae in the operations in ℚL and vice versa, a homomorphism of one object is also a homomorphism of the other (being the same mapping).

In view of the connection between the group and Lie ring operations, we shall freely use any of the notations for ng = g^{n}, where g ∈ F, n ∈ ℤ, since this cannot lead to any confusion; for example, then nF is the set of all nth powers of elements of F. Recall that F^{n} denotes the subgroup of F generated by all nth powers, that is, F^{n} = ⟨nF ⟩. Extending this notation we have also ℚF = F̂.

We shall say that a value is (a, b, …)-bounded if it is bounded in terms of a, b, … only.

Lemma 1. There exist c-bounded positive integers n_{1} = n_{1}(c) and n_{2} = n_{2}(c) such that n_{1}L ⊆ F ⊆ n_{2}^{−1}L.

Proof. First we prove that F ⊆ n_{2}^{−1}L. For an arbitrary k the product x_{1}x_{2} ⋯ x_{k}, being an element of ℚL, is a rational linear combination of Lie ring commutators in the x_{i}. Any commutator in this combination involves at most c elements x_{i_{1}}, …, x_{i_{d}}, d ≤ c, since the algebra is nilpotent of class c. It remains intact under the homomorphism sending x_{i_{j}} to x_{i_{j}}, j = 1, …, d, and all other free generators to 0. Note that this is also a homomorphism of F̂ (recall that ℚL ∋ 0 = 1 ∈ F̂). Hence the coefficient at our commutator is the same as in the similar expansion of the product x_{i_{1}} ⋯ x_{i_{d}}. Since d ≤ c, the latter obviously belongs to n_{2}^{−1}L for some c-bounded positive integer n_{2} = n_{2}(c). Hence x_{1}x_{2} ⋯ x_{k} ∈ n_{2}^{−1} ⟨x_{1}, …, x_{k}⟩_{L} for any k. Then any element of F, having the form ∏ x_{j_{i}}^{r_{i}} for r_{i} ∈ ℤ, belongs to n_{2}^{−1} ⟨r_{1}x_{j_{1}}, r_{2}x_{j_{2}}, … ⟩_{L} ⊆ n_{2}^{−1}L.

We now prove that n_{1}L ⊆ F. We know that (x_{1} + ⋯ + x_{c})^{f_{1}} ∈ F for some positive integer f_{1} = f_{1}(c), since x_{1} + ⋯ + x_{c} ∈ ℚL = F̂. Then also (x_{i_{1}} + ⋯ + x_{i_{d}})^{f_{1}} ∈ F for any d ≤ c. We claim that (x_{1}+⋯+x_{k})^{f_{1}} ∈ F for any k ∈ ℕ for the same exponent f_{1}. Indeed, (x_{1} + ⋯ + x_{k})^{f_{1}} = ∏ c_{j}^{r_{j}} is a product of rational powers of basic (group) commutators c_{j} in the x_{i} in some fixed order. We need to show that r_{j} ∈ ℤ for all j. Any commutator c_{t} depends only on at most c variables x_{i_{1}}, …, x_{i_{d}}, d ≤ c. Applying the homomorphism sending x_{i_{k}} to x_{i_{k}} for k = 1, …, d and the other generators to 1 (that is, to 0 ∈ ℚL), we get the expression of (x_{i_{1}} + ⋯ + x_{i_{d}})^{f_{1}} as the image of ∏ c_{j}^{r_{j}}, in which the factor c_{t}^{r_{t}} remains intact. Since this is an element of F by the choice of f_{1}, we must have r_{t} ∈ ℤ, as required. Next, there is a positive integer f_{2} = f_{2}(c) such that f_{2}[x_{i_{1}}, …, x_{i_{d}}]_{L} ∈ F for any d ≤ c. Now any element of L has the form ∑ α_{i}c_{i}, where α_{i} ∈ ℤ and the c_{i} are (simple) commutators of weight ≤ c in the x_{j}. Applying the homomorphism extending the map x_{i} → α_{i}f_{2}c_{i} to the inclusion f_{1} ∑ x_{i} ∈ ⟨x_{1}, x_{2} … ⟩_{G} we derive that f_{1}f_{2} ∑ α_{i}c_{i} = f_{1} ∑ f_{2}α_{i}c_{i} ∈ ⟨f_{2}α_{1}c_{1}, f_{2}α_{2}c_{2}, … ⟩_{G}. The right-hand side is contained in F since f_{2}α_{i}c_{i} ∈ F for all i. Thus we can put n_{1} = f_{1}f_{2}. □

Lemma 2. There are c-bounded positive integers n_{3} = n_{3}(c) and n_{4} = n_{4}(c) such that nn_{3}L ⊆ F^{n} ⊆ nn_{4}^{−1}L for any n ∈ ℕ.

Proof. Since F^{n} ⊇ nF ⊇ nn_{1}L by Lemma 1, the first inclusion holds with n_{3} = n_{1}. We now prove the second inclusion. For any k ∈ ℕ we have x_{1} ⋯ x_{k} ∈ n_{2}^{−1}L by Lemma 1. For any g_{i} ∈ F the homomorphism extending the map x_{i} → ng_{i} (= g_{i}^{n}) gives that g_{1}^{n} ⋯ g_{k}^{n} ∈ n_{2}^{−1}S, where S = ⟨ng_{1}, …, ng_{k}⟩_{L}. Obviously, S ⊆ nS_{1}, where S_{1} = ⟨g_{1}, …, g_{k}⟩_{L}. Again g_{i} ∈ n_{2}^{−1}L by Lemma 1; since the nilpotency class is c, we have S_{1} ⊆ n_{2}^{−c}L. Hence F^{n} ⊆ nn_{2}^{−c−1}L. Thus we can set n_{4} = n_{2}^{c+1}. □

Here and later we use the fact that the additive group of ℚL is torsion-free and hence has unique roots, so that rS ⊆ T ⇔ S ⊆ r^{−1}T for any r ∈ ℚ and any subsets S, T ⊆ ℚL. The following lemma is a generalization of Blackburn’s lemma from [2].

Lemma 3. There is a c-bounded positive integer n_{5} = n_{5}(c) such that for any n divisible by n_{5} in any nilpotent group of class c every product of nth powers of elements is a nn_{5}^{−1}th power of some element.

Proof. It is sufficient to prove the lemma for F; then a suitable homomorphism F → G gives the result for G. By Lemma 2 we have F^{n} ⊆ nn_{4}^{−1}L ⊆ nn_{4}^{−1}n_{3}^{−1}F, so that we can put n_{5} = n_{3}n_{4}. □

Lemma 4. For n_{5} from Lemma 3, for any n divisible by n_{5} the element [x_{1}^{n}, x_{2}, …, x_{k}]_{G} is a nn_{5}^{−1}th power of a product of integer powers of group commutators in x_{1}, …, x_{k}, each involving all these elements.

Proof. Since [x_{1}^{n}, x_{2}, …, x_{k}]_{G} ∈ F^{n}, by Lemma 3 we have already [x_{1}^{n}, x_{2}, …, x_{k}]_{G} = (∏ c_{i}^{r_{i}})^{nn_{5}^{−1}} or, equivalently, n_{5}n^{−1}[x_{1}^{n}, x_{2}, …, x_{k}]_{G} = ∏ c_{i}^{r_{i}}, where r_{i} ∈ ℤ for all i and the c_{i} are basic commutators in the x_{j} in some fixed order. We need to show that only those commutators occur (non-trivially) that involve all elements x_{1}, …, x_{k}. To show that all c_{i} do depend on a given x_{t}, t ∈ {1, 2, …, k}, we apply the homomorphism extending the map x_{t} → 1, x_{j} → x_{j} for j ≠ t to the above equality. The left-hand side becomes 1 = 0 ∈ ℚL. On the right all factors involving x_{t} vanish, while all without x_{t} remain intact. By the uniqueness of a presentation in the form of such a product we must have only trivial powers of those c_{i} that do not depend on x_{t}. □

Lemma 5. There exists a c-bounded positive integer n_{6} = n_{6}(c) such that for any n divisible by n_{6}

x_{1}^{n} + x_{2} = x_{1}^{n} · x_{2} · A(x_{1}, x_{2}),  (1)

where A(x_{1}, x_{2}) ∈ ⟨[x_{1}^{nn_{6}^{−1}}, x_{2}]_{G}⟩^{F};

[x_{1}^{n}, x_{2}]_{L} = [x_{1}^{n}, x_{2}]_{G} · B(x_{1}, x_{2}),  (2)

where B(x_{1}, x_{2}) ∈ ⟨[[x_{1}^{nn_{6}^{−1}}, x_{2}]_{G}, ⟨x_{1}^{n}, x_{2}⟩]_{G}⟩^{F};

x_{1}^{n} · x_{2} = x_{1}^{n} + x_{2} + C(x_{1}, x_{2}),  (3)

where C(x_{1}, x_{2}) belongs to the ideal generated by [nn_{6}^{−1}x_{1}, x_{2}]_{L} in ⟨x_{1}, x_{2}⟩_{L};

[x_{1}^{n}, x_{2}]_{G} = [x_{1}^{n}, x_{2}]_{L} + D(x_{1}, x_{2}),  (4)

where D(x_{1}, x_{2}) belongs to the ideal generated by [[nn_{6}^{−1}x_{1}, x_{2}]_{L}, ⟨nx_{1}, x_{2}⟩_{L}]_{L} in ⟨x_{1}, x_{2}⟩_{L}.

(Of course, it is simply to save notation that we choose the same number n_{6} for all four equations, the least common multiple, say.)

Proof. By the inversions of the Baker–Hausdorff formula, A(x_{1}, x_{2}) in (1) is a product of some rational powers of commutators in x_{1}^{n} and x_{2}. Since we do not need uniqueness, without loss we may assume that only simple commutators in x_{1}^{n} and x_{2} occur, each starting with [x_{1}^{n}, x_{2}]. Let f(c) be the least common multiple of the denominators of the exponents, which depends only on c. For each factor we apply Lemma 4 using the first occurrence of x_{1}^{n} in the commutator. For n_{6} divisible by f(c)n_{5}(c) the commutator is a f(c)th power of a product of integer powers of commutators in x_{1}^{nn_{6}^{−1}}, x_{2}. Hence each factor is such a product too, so that A(x_{1}, x_{2}) ∈ ⟨[x_{1}^{nn_{6}^{−1}}, x_{2}]_{G}⟩^{F}.

Similarly, B(x_{1}, x_{2}) in (2) is a product of rational powers of simple commutators of weight ≥ 3 in x_{1}^{n} and x_{2} beginning with [x_{1}^{n}, x_{2}, y]_{G}, where y is either x_{1}^{n} or x_{2}. We get the required result in the same way as for (1), applying Lemma 4 to each commutator using the first entry of x_{1}^{n}.

By the Baker–Hausdorff formula, C(x_{1}, x_{2}) in (3) is a rational linear combination of Lie ring commutators in x_{1}^{n} = nx_{1} and x_{2}. We may assume them all to be simple and to begin with [nx_{1}, x_{2}]_{L}. It suffices to take for n_{6} the least common multiple of the denominators of the coefficients, which depends only on c.

Similarly, D(x_{1}, x_{2}) in (4) is a linear combination of simple Lie ring commutators of weight ≥ 3 in x_{1}^{n} = nx_{1} and x_{2} beginning with [nx_{1}, x_{2}, y]_{L}, where y is either nx_{1} or x_{2}. Again it suffices to take n_{6} to be the least common multiple of the denominators of the coefficients, which depends only on c. □

Lemma 6. There are c-bounded positive integers n_{7} = n_{7}(c) and n_{8} = n_{8}(c) such that for any n divisible by n_{8} we have g + nn_{7}L ⊆ g · F^{n} ⊆ g + nn_{8}^{−1}L for any g ∈ F.

Proof. For any l ∈ L by (1) we have nn_{6}n_{3}l+g = nn_{6}n_{3}l·g·A, where A ∈ [nn_{3}l, F]_{G} ≤ F^{n} since nn_{3}l ∈ F^{n} by Lemma 2. Since also nn_{6}n_{3}l ∈ F^{n}, we have nn_{6}n_{3}l + g ∈ g · F^{n}. Thus, we can put n_{7} = n_{6}n_{3}.

By Lemma 2 we have F^{n} ⊆ nn_{4}^{−1}L. Hence any element of g · F^{n} = F^{n}· g has the form nn_{4}^{−1}l·g for l ∈ L. By (3) for n divisible by n_{4}n_{6} we have nn_{4}^{−1}l·g = nn_{4}^{−1}l+g+C, where C is in the ideal generated by [nn_{4}^{−1}n_{6}^{−1}l, g]_{L} in ⟨l, g⟩_{L}. Since g ∈ n_{2}^{−1}L by Lemma 1 and the nilpotency class is c, we have C ∈ nn_{4}^{−1}n_{6}^{−1}n_{2}^{−c+1}L whence g · F^{n} ⊆ g + nn_{4}^{−1}n_{6}^{−1}n_{2}^{−c+1}L. Thus we can put n_{8} = n_{4}n_{6}n_{2}^{c−1}. □

Lemma 7. For any n ∈ ℕ we have F^{nn_{3}n_{4}n_{6}} ⊆ n_{6}F^{n}.

Proof. By Lemma 2 we obtain F^{nn_{3}n_{4}n_{6}} ⊆ n_{6}nn_{3}L ⊆ n_{6}F^{n}. □

Remark. It is possible to give effective bounds for the functions n_{i}(c) in the lemmas above (and in the theorems from the next sections). For example, let us estimate n_{2} from Lemma 1. Let A be the free nilpotent of class c associative ℤ-ring with 1 on variables x_{i} naturally embedded into the ℚ-algebra ℚA. The elements e^{x_{1}}, e^{x_{2}} … freely generate a free nilpotent group F of class c. The free nilpotent Lie algebra ℚL of class c can be identified with the Lie algebra generated by the x_{i} with respect to the operation [a, b] = ab − bam and the Lie ring L with the Lie generated by the x_{i}. Induction on the length of a product in the e^{±x_{i_{j}}} shows that F ⊆ \frac{1}{c!}A. Then log f ∈ \frac{1}{(c!)^{2}} A. By the Baker–Hausdorff formula log f ∈ ℚL Since ℚL∩ \frac{1}{(c!)^{2}} A = \frac{1}{(c!)^{2}} L, we see that log f ∈ \frac{1}{(c!)^{2}} L. Tracing the proof of Lemma 1 we see that one can put n_{2} = (c!)^{2}. Other functions n_{i} can be estimated basing on this information on the Baker–Hausdorff formula and the estimate for n_{2}. In particular, a rather crude induction by c gives that one can put n_{1} = n_{2}^{c}, but there must be a much better, “more symmetric” estimate for n_{1}, probably involving some better estimates for the inverses of the Baker–Hausdorff formula. It may be interesting to find better (or even the best) estimates for the functions n_{i}. Note that in applications to nilpotent p-groups, it is only p-adic estimates that matter.

2. A correspondence between nilpotent Lie rings and nilpotent groups

First we proceed with our construction for a free nilpotent group; then the same result will follow for an arbitrary nilpotent group.

Theorem 1. There is a c-bounded positive integer n_{9} = n_{9}(c) such that for any n divisible by n_{9} the following holds.

(a) The subgroup U = F^{n} is a Lie ℤ-subring of L (that is, U is closed under the operations ± and [,]_{L}).

(b) Any normal subgroup N of F which is contained in U is an ideal of the Lie ring U and g + N = g · N for any g ∈ U.

Let U_{1}, N_{1}, and N_{2} be normal subgroups of F such that U ⊇ U_{1} ⊇ N_{1} ⊇ N_{2}.

(c) If the factor-group N_{1}/N_{2} is abelian, then the Lie factor-ring N_{1}/N_{2} is abelian.

(d) If the factor-group N_{1}/N_{2} is central in the group U_{1}, then the Lie factor-ring N_{1}/N_{2} is central in the Lie ring U_{1}.

(e) If I is an ideal of U contained in V = F^{nn_{3}n_{4}n_{6}}, then the same set I is a normal subgroup of the group F^{nn_{3}n_{4}n_{6}}.

Let V_{1}, I_{1}, and I_{2} be ideals of U such that V ⊇ V_{1} ⊇ I_{1} ⊇ I_{2}.

(f) If the Lie factor-ring I_{1}/I_{2} is abelian, then the factor-group I_{1}/I_{2} is abelian.

(g) If the Lie factor-ring I_{1}/I_{2} is central in V_{1}, then the factor-group I_{1}/I_{2} is central in the group V_{1}.

Remark. The parts (e)–(g) of the theorem, with passages from Lie rings to groups, do not seem quite dual to the parts (a)–(d), with passages from groups to Lie rings. In fact, the corresponding dual statements do hold true (we leave them as an exercise to the reader). For example, for n divisible by some n_{10}(c), if J is an ideal of L contained in nL then the same set J is a normal subgroup of the group nL ≤ F. But it is statements (e)–(g) of Theorem 1 that we shall need in our applications to problems about groups; apparently, there is no Lie ring structure associated beforehand with an arbitrary nilpotent group G, and applying a homomorphism of F onto G we have to rely on subgroups.

Proof. We assume in what follows that n is divisible by n_{4}n_{6}n_{5}; then U = F^{n} ⊆ n_{6}F ∩L by Lemmas 2 and 3.

(a) Let a, b ∈ U. We have a = a_{1}^{n_{6}} for some a_{1} ∈ F. Then by (1) we have a+b = a·b·A, where A ∈ [F, b]_{G} ≤ U, since b ∈ U ⊲_{G} F. By (2) we have [a, b]_{L} = [a, b]_{G} · B, where B ∈ [F, b]_{G} ≤ U for the same reason.

(b) Let a, b ∈ N. Again a = a_{1}^{n_{6}} for some a_{1} ∈ F. By (1) we have a + b = a · b · A, where A ∈ [F, b]_{G} ≤ N since b ∈ N ⊲_{G} F. For any g ∈ U we have also g = g_{1}^{n_{6}} for some g_{1} ∈ F. Then by (2) we have [g, b]_{L} = [g, b]_{G} · B, where B ∈ [F, b]_{G} ≤ N for the same reason. Next, by (1) we have g + b = g · b · A, where A ∈ [F, b]_{G} ≤ N, so that g + N ⊆ gN. Now let n be divisible by n_{6}n_{5}^{2}. Then g = g_{1}^{n_{6}} for some g_{1} ∈ F^{n_{6}n_{5}}. By (3) we have g · b = g + b + C(g_{1}, b), where C(g_{1}, b) is in the ideal generated by [g_{1}, b]_{L} in ⟨g_{1}, b⟩_{L}. But we have shown above that N ⊲_{L} F^{n_{6}n_{5}}. Since b ∈ N, we have C(g_{1}, b) ∈ N, so that g · N ⊆ g + N.

(c) Let a, b ∈ N_{1}. We have a = a_{1}^{n_{6}} for some a_{1} ∈ F. By (2) we have [a, b]_{L} = [a, b]_{G}·B, where B is in the normal closure in F of [[a_{1}, b]_{G}, ⟨a, b⟩_{G}]_{G}. But [a_{1}, b]_{G} ∈ N_{1} since b ∈ N_{1} ⊲_{G} F; also ⟨a, b⟩_{G} ≤ N_{1}. Hence [[a_{1}, b]_{G}, ⟨a, b⟩_{G}]_{G} ≤ [N_{1}, N_{1}] ≤ N_{2} ⊲_{G} F, whence [a, b]_{L} ∈ N_{2}.

(d) Let g ∈ U_{1} and b ∈ N_{1}. We have g = g_{1}^{n_{6}} for some g_{1} ∈ F. By (2) we have [g, b]_{L} = [g, b]_{G} · B, where B is in the normal closure in F of [[g_{1}, b]_{G}, ⟨g, b⟩_{G}]_{G}. But [g_{1}, b]_{G} ∈ N_{1} since b ∈ N_{1} ⊲_{G} F, and ⟨g, b⟩_{G} ≤ U_{1}. Hence [[g_{1}, b]_{G}, ⟨g, b⟩_{G}]_{G} ≤ [N_{1}, U_{1}] ≤ N_{2} ⊲_{G} F, whence [g, b]_{L} ∈ N_{2}.

(e) Let v, w ∈ I. Then v = n_{6}v_{1} for some v_{1} ∈ U by Lemma 7. By (3) then v · w = v + w + C, where C belongs to the ideal generated by [v_{1}, w]_{L} in ⟨v_{1}, w⟩_{L}. This ideal is contained in I since w ∈ I ⊲_{L} U and v_{1} ∈ U. Hence v · w ∈ I. Since v^{−1} = −v ∈ I too, I is a subgroup. For any g ∈ V by Lemma 7 we have g = g_{1}^{n_{6}} for some g_{1} ∈ U. By (4) then [g, w]_{G} = [g, w]_{L} + D, where D is in the ideal generated by [[g_{1}, w]_{L}, ⟨g, w⟩_{L}]_{L} in ⟨g_{1}, w⟩_{L}. This ideal is contained in I since [g_{1}, w]_{L} ∈ I ⊲_{L} U and ⟨g, w⟩_{L} ≤ U. Thus, I is a normal subgroup of V.

(f) Let v, w ∈ I_{1}. Then v = n_{6}v_{1} for some v_{1} ∈ U by Lemma 7. By (4) then [v, w]_{G} = [v, w]_{L} + D, where D is in the ideal generated by [[v_{1}, w]_{L}, ⟨v, w⟩_{L}]_{L} in ⟨v_{1}, w⟩_{L}. This ideal is contained in [I_{1}, I_{1}]_{L} ≤ I_{2} since [v_{1}, w]_{L} ∈ I_{1} and ⟨v, w⟩_{L} ≤ I_{1}; also [v, w]_{L} ∈ [I_{1}, I_{1}]_{L} ≤ I_{2}. Hence [v, w]_{G} ∈ I_{2}.

(g) Let w ∈ I_{1} and g ∈ V_{1}. Then g = n_{6}g_{1} for some g_{1} ∈ U by Lemma 7. By (4) then [g, w]_{G} = [g, w]_{L} + D, where D is in the ideal generated by [[g_{1}, w]_{L}, ⟨g, w⟩_{L}]_{L} in ⟨g_{1}, w⟩_{L}. Since [g_{1}, w]_{L} ∈ I_{1} and ⟨g, w⟩_{L} ≤ V_{1}, this ideal is contained in [I_{1}, V_{1}] ≤ I_{2}; also [g, w]_{L} ∈ [V_{1}, I_{1}]_{L} ≤ I_{2}. Hence [g, w]_{G} ∈ I_{2}. □

Now suppose that P is an arbitrary nilpotent group of class c. Let F be a free nilpotent group of class c constructed within ℚL as above on a sufficiently large set of free generators, and let π : F → P be an epimorphism of F onto P. Then π(F^{n}) = P^{n}. For n_{9} from Theorem 1, for any n divisible by n_{9} we now define the new operations on the set P^{n} as follows:

π(a) + π(b) = π(a + b); [π(a), π(b)]_{L} = π([a, b]_{L})  (5)

for any a, b ∈ F^{p^{n}}. Let us show that this definition is correct. Let N = Ker π ∩ F^{n} ⊲ F; then N ⊲_{L} F^{n} by Theorem 1(b). Suppose that a′N = aN and b′N = bN. Then

(a′ + b′)N = (a′ + b′) + N = (a′ + N) + (b′ + N) = a′N + b′N =
= aN + bN = (a + N) + (b + N) = (a + b) + N = (a + b)N

by Theorem 1(b). Further, by Theorem 1(b),

[a′, b′]_{L}N = [a′, b′]_{L} + N = [a′ + N, b′ + N]_{L} = [a′N, b′N]_{L} =
= [aN, bN]_{L} = [a + N, b + N]_{L} = [a, b]_{L} + N = [a, b]_{L}N.

Theorem 2. (a) The set \overline{U} = P^{n} is a Lie ring with respect to operations (5).

(b) Any normal subgroup \overline{N} of P which is contained in \overline{U} is an ideal of the Lie ring \overline{U} and g + \overline{N} = g\overline{N} for any g ∈ \overline{U}.

Let \overline{U}_{1}, \overline{N}_{1}, and \overline{N}_{2} be normal subgroups of P such that \overline{U} ⊇ \overline{U}_{1} ⊇ \overline{N}_{1} ⊇ \overline{N}_{2}.

(c) If the factor-group \overline{N}_{1}/\overline{N}_{2} is abelian, then the Lie factor-ring \overline{N}_{1}/\overline{N}_{2} is abelian. The Lie ring \overline{U} is soluble of derived length not greater than the derived length of the group \overline{U}.

(d) If the factor-group \overline{N}_{1}/\overline{N}_{2} is central in the group \overline{U}_{1}, then the Lie factor-ring \overline{N}_{1}/\overline{N}_{2} is central in the Lie ring \overline{U}_{1}. The Lie ring \overline{U}_{1} is nilpotent of class not greater than the nilpotency class of the group \overline{U}_{1}.

(e) If \overline{I} is an ideal of \overline{U} contained in \overline{V} = P^{nn_{3}n_{4}n_{6}}, then the same set \overline{I} is a normal subgroup of the group P^{nn_{3}n_{4}n_{6}}.

Let \overline{V}_{1}, \overline{I}_{1}, and \overline{I}_{2} be ideals of \overline{U} such that \overline{V} ⊇ \overline{V}_{1} ⊇ \overline{I}_{1} ⊇ \overline{I}_{2}.

(f) If the Lie factor-ring \overline{I}_{1}/\overline{I}_{2} is abelian, then the factor-group \overline{I}_{1}/\overline{I}_{2} is abelian. The derived length of the subgroup P^{nn_{3}n_{4}n_{6}} is equal to the derived length of the Lie ring \overline{V}.

(g) If the Lie factor-ring \overline{I}_{1}/\overline{I}_{2} is central in \overline{V}_{1}, then the factor-group \overline{I}_{1}/\overline{I}_{2} is central in the group \overline{V}_{1}. The nilpotency class of the subgroup \overline{V}_{1} is equal to the nilpotency class of the Lie ring \overline{V}_{1}.

(h) If φ is an automorphism of P, then the restriction of φ to \overline{U} is a Lie ring automorphism of \overline{U}.

Proof. We may assume that \overline{U} = π(U) with U as in Theorem 1, \overline{N} = π(N), and so on. Then the statements (a)–(g) follow directly from Theorem 1 by the definition of the operations (5), because they all can be expressed either by formulae or by inclusions. For example, since [a, b, c]_{L} + [b, c, a]_{L} + [c, a, b]_{L} = 0 in U, applying π we obtain the Jacobi identity in \overline{U}:

[π(a), π(b), π(c)]_{L} + [π(b), π(c), π(a)]_{L} + [π(c), π(a), π(b)]_{L} = 0.

Another example: if [\overline{N}_{1}, \overline{N}_{1}]_{G} ≤ \overline{N}_{2} in \overline{U}, then [N_{1}, N_{1}]_{G} ≤ N_{2} in U and hence, by Theorem 1(c), [N_{1}, N_{1}]_{L} ≤ N_{2}; applying π we obtain that [\overline{N}_{1}, \overline{N}_{1}]_{L} ≤ \overline{N}_{2} too. To obtain the additional statements about the derived length and the nilpotency class one only needs to notice that the terms of the derived series and the lower central series in the subgroup \overline{U}_{1} are normal in P, and that such terms in the Lie ring \overline{V}_{1} are ideals in \overline{U}.

Let us prove (h). For given c the polynomials A(x_{1}, x_{2}) and B(x_{1}, x_{2}) in Lemma 5 can be regarded as fixed; then the same formulae hold for arbitrary elements in F. By Lemma 3 for any x ∈ F^{n} there is x′ ∈ F such that x = x′^{n_{6}}. Then π(x + y) = π(xyA(x′, y)) = π(x)π(y)A(π(x′), π(y)). Hence in \overline{U} we have u + v = uvA(u′, v) whenever u = u′^{n_{6}} and such u′ always exists. Now we have φ(u + v) = φ(uvA(u′, v)) = φ(u)φ(v)A(φ(u′), φ(v)) = φ(u)+φ(v). Similarly, [u, v]_{L} = [u, v]_{G}B(u′, v) for any u, v ∈ \overline{U} if u = u′^{n_{6}}; hence φ([u, v]_{L}) = [φ(u), φ(v)]_{G}B(φ(u′), φ(v)) = [φ(u), φ(v)]_{L}. □

Remark. Suppose that P is a nilpotent p-group of class c, for some prime p. Then for any k coprime to p it contains the unique kth root of any g ∈ P; in fact, this root is in ⟨g⟩. (In other words, any p-group is a ℚ_{σ}-group for σ being the set of all primes other than p.) Then any subgroup of the form P^{n} is actually equal to P^{p^{m}}, where p^{m} is the highest power of p dividing n. The additive groups of the Lie rings \overline{U} and \overline{V} constructed in Theorem 2 are also p-groups (so that these are Lie ℚ_{σ}-algebras), since for commuting elements the addition in these Lie rings coincides with the group multiplication. The function n_{9}(c) here can be replaced by its refinement p^{m(p,c)}, where m(p, c) depends on p as well as on c. The value of this function m(p, c) is the better (that is, smaller) the larger is p. In particular, if p > c, one can put m(p, c) = 0, which amounts to the Lazard correspondence starting working, when the group P itself becomes a Lie ℚ_{σ}-algebra.

3. Reductions to Lie rings

Suppose that a finite p-group G admits an automorphism φ of order p having exactly p^{m} fixed points. By Khukhro’s theorem [6] then G has a subgroup of (p, m)-bounded index which is nilpotent of p-bounded class. (The bound for the nilpotency class was actually h(p) + 1 in [6] and it was improved to h(p) by Makarenko [12], where h(p) is Higman’s function bounding the nilpotency class of a Lie ring with a fixed-point-free automorphism of order p.) On the other hand Medvedev [15] proved that G has also a subgroup of (p, m)-bounded index which is nilpotent of m-bounded class ≤ g(m). This result follows from his theorem on Lie rings: if a Lie ring with additive p-group admits an automorphism of order p having exactly p^{m} fixed points, then it has a subring of (p, m)-bounded index (in the additive group) which is nilpotent of m-bounded class ≤ l(m). The result for p-groups is obtained by reduction to Lie rings, which can be performed either via the Lie rings for uniformly powerful p-groups, or using the Lazard correspondence and the above theorem giving a subgroup of bounded index of nilpotency class ≤ h(p). But in both cases the function g(m) for groups becomes much larger than the function l(m) for Lie rings. (Refining the arguments from [14] and [16] based on uniformly powerful p-groups, one can achieve g(m) = 4ml(m), see [8, Chapter 14]. Reduction by the Lazard correspondence gives the same value g(m) = l(m), but only for sufficiently large primes p ≥ f(m).)

Now we use the correspondence from Section 2 to show that one can always take g(m) = l(m). Here we do not care much about the index of the ambient subgroup/subring, provided it remains (p, m)-bounded.

Theorem 3. Let l(m) be Medvedev’s function for Lie rings, that is, such that if a Lie ring whose additive group is a p-group admits an automorphism of order p having exactly p^{m} fixed points, then it has a subring of (p, m)-bounded index (in the additive group) which is nilpotent of class ≤ l(m). Then any finite p-group P admitting an automorphism φ of order p with exactly p^{m} fixed points has a subgroup of (p, m)-bounded index which is nilpotent of class ≤ g(m) = l(m).

Proof. By the above we may assume P to be nilpotent of class c ≤ h(p). Then for some p-bounded positive integer n = n(c) the set P^{p^{n}} can be endowed with the structure of a Lie ring U with the subring V = P^{p^{n+n′}} satisfying Theorem 2, where p^{n′} is the highest power of p dividing n_{3}n_{4}n_{6}. By Medvedev [15] for some (p, m)-bounded positive integer f = f(p, m) the ideal p^{f}V is nilpotent of class ≤ l(m). Since this is also an ideal of U, by Theorem 2(e) this set is a normal subgroup of P^{p^{n}} and hence coincides with the subgroup V^{p^{f}}. By Theorem 2(g) it has the same nilpotency class ≤ l(m). Since the ranks of all φ-invariant sections of P are bounded by mp (see, for example, [8, Corollary 2.7]), this subgroup has (p, m)-bounded index in P. □

Another application is a new version of reduction to Lie rings in the proof of the following theorem of Khukhro [7]: if a finite p-group P admits an automorphism of order p^{n} with exactly p^{m} fixed points, then P has a subgroup of (p, m, n)-bounded index which is soluble of derived length ≤ 2k, where k = k(p^{n}) is Kreknin’s function bounding the derived length of a Lie ring with a fixed-point-free automorphism of order p^{n}. In the first part of the proof in [7] it is shown that the nilpotency class of P_{1}^{(k)} is (p, m, n)-bounded for some subgroup P_{1} of (p, m, n)-bounded index. The Mal’cev correspondence is applied to a free nilpotent φ-group and the result is then translated into the existence of a subgroup P_{2} of (p, m, n)-bounded index in P_{1}^{(k)} such that P_{2} has derived length ≤ k. Via the properties of powerful p-groups this produces the required result.

We now show how the correspondence from Section 2 can be applied to find that subgroup P_{2} in H = P_{1}^{(k)}. Since H is nilpotent of (p, m, n)-bounded class c, for some (p, m, n)-bounded positive integer f = f(c) the set H^{p^{f}} can be endowed with the structure of a Lie ring U with the subring V = H^{p^{f+f′}} satisfying Theorem 2, where p^{f′} is the highest power of p dividing n_{3}n_{4}n_{6}. By Kreknin’s theorem p^{m}(p^{n}V)^{(k)} = 0 and hence, the more so, (p^{n+m}V)^{(k)} = 0. Since p^{n+m}V is an ideal of U, by Theorem 2(e) the set p^{n+m}V is a normal subgroup of H which coincides with V^{p^{n+m}}. By Theorem 2(f) this subgroup is soluble of the same derived length ≤ k. Since the ranks of all φ-invariant sections are bounded by mp^{n} (see, for example, [8, Corollary 2.7]), this subgroup has (p, m, n)-bounded index in H.

Although this argument does not improve the bound for the derived length in [7] (which is virtually best-possible, see the next section), it provides another illustration of application of our correspondence from Section 2.

4. Examples with Higman’s and Kreknin’s functions

Higman’s [4] and Kreknin’s [9] theorems are essentially about graded Lie rings, while applications to Lie rings with fixed-point-free automorphisms of finite order are based on eigen-space decomposition. There may be, however, a certain difference between these functions for Lie rings (ℤ-algebras), h_{ℤ} and k_{ℤ}, and for Lie algebras over a field, over ℚ, say, denoted by h_{ℚ} and k_{ℚ}.

We define Higman’s function h_{ℤ}(p) as follows. For a prime p let L = L_{0} ⊕ L_{1} ⊕ ⋯ ⊕ L_{p−1} be a free (ℤ/pℤ)-graded Lie ring (ℤ-algebra) with [L_{i}, L_{j}] ⊆ L_{i+j (mod p)}. Higman’s theorem states that L/_{id}⟨L_{0}⟩ is nilpotent of p-bounded class h_{ℤ}(p), which defines Higman’s function. It follows that if a Lie ring M admits a fixed-point-free automorphism φ of order p then M is nilpotent of class ≤ h_{ℤ}(p).

A similar definition based on a free (ℤ/pℤ)-graded Lie ℚ-algebra ℚL (where L is as above) may, in principle, give a smaller value h_{ℚ}(p) for the nilpotency class of ℚL/_{id}⟨ℚL_{0}⟩. This may happen if the additive group of L/_{id}⟨L_{0}⟩ has a large torsion subgroup containing some non-trivial terms of the lower central series of this quotient. It is yet unclear whether there can really be such torsion subgroup, nor it is known if h_{ℚ}(p) = h_{ℤ}(p) for all primes. We only know that there is no such torsion for p = 2 and p = 3, that h_{ℚ}(5) = 6 = h_{ℤ}(5), due to Higman [4], and that h_{ℚ}(7) = 12 = h_{ℤ}(7), due to Higman’s examples [4] showing that h_{ℚ}(p) ≥ (p^{2} − 1)/4 for all primes and calculations by Scimemi (unpublished) and Hughes [5] who showed that h_{ℤ}(7) ≤ 12. It is also worth to mention that an explicit upper

estimate for h_{ℤ}(p), which is due to Kreknin and Kostrikin [9], [10], is \frac{(p − 1)^{2^{p−1}−1} − 1}{p − 2}.

Note that for each prime p there exists a Lie ℚ-algebra of class exactly h_{ℚ}(p) with a fixed-point-free automorphism φ of order p: simply define the action of φ on the image of the component ℚL_{j} in ℚL/_{id}⟨ℚL_{0}⟩ as multiplication by ω^{j}, where ω is a primitive pth root of unity. But for h_{ℤ} we do not know whether p-torsion in L/_{id}⟨L_{0}⟩ can prevent one from constructing examples of nilpotent Lie rings of nilpotency class exactly h_{ℤ}(p) with a fixed-point free automorphism of order p.

Recall that if a finite p-group admits an automorphism of order p with exactly p^{m} fixed points then it has a subgroup of (p, m)-bounded index which is nilpotent of class ≤ h(p). Although it was function h(p) = h_{ℤ}(p) that was used in the papers [1], [6], [12], in fact, one can prove the same kind of result with h(p) = h_{ℚ}(p) (at the expense of a possibly worse (p, m)-bounded estimate for the index of the subgroup). Indeed, it is the inclusion p^{h(p)+1}[x_{1}, …, x_{h(p)+1}] ∈ _{id}⟨C_{L}(φ)⟩ for a Lie ring L admitting an automorphism φ of order p that was used in the proofs. In fact, it is sufficient to have in this situation just p^{f(p,m)}[x_{1}, …, x_{h(p)+1}] ∈ _{id}⟨C_{L}(φ)⟩ for some (p, m)-bounded positive integer f(p, m). From the definition of h = h_{ℚ}(p) it follows that in a free graded Lie algebra the commutator [x_{1}, …, x_{h+1}] is a rational linear combination of commutators in the grading components of the elements x_{i} that have subcommutators in L_{0}. Hence there is even a p-bounded positive integer α(p) such that p^{α(p)}[x_{1}, …, x_{h+1}] is a p-integer linear combination of the same commutators. Because of the inclusion pL ⊆ L_{0} + ⋯ + L_{p−1}, where L_{j} are analogues of eigen-spaces, which holds for any Lie ring L with an automorphism φ of order p, now we have p^{α(p)+h+1}[x_{1}, …, x_{h+1}] ∈ _{id}⟨C_{L}(φ)⟩. Applying the rest of the arguments from [6] and [12], we obtain the required result.

Theorem 4. If a finite p-group P admits an automorphism of order p with exactly p^{m} fixed points then P has a subgroup of (p, m)-bounded index which is nilpotent of class ≤ h_{ℚ}(p). □

We shall now show that Higman’s function h_{ℚ}(p) for Lie ℚ-algebras is the best possible bound for the nilpotency class in this group-theoretic result (if required to depend on p only).

As noted above there is a nilpotent Lie ℚ-algebra N of class exactly h = h_{ℚ}(p) with a fixed-point-free automorphism φ of order p. There are elements n_{1}, …, n_{h} ∈ N such that [n_{1}, …, n_{h}] ≠ 0. We assume that N = ⟨n_{i}^{φ^{j}}⟩. By the Mal’cev correspondence we can view N as a nilpotent ℚ-group and φ as its automorphism acting on the same set in the same way.

Let σ be the set of all primes other than p, and let ℚ_{σ} = {m/n | m, n ∈ ℤ, p ∤ n} be the ring of p-integers. Let M be the Lie ℚ_{σ}-subring of N generated by {n_{i}^{φ^{j}} | i = 1, …, h; j = 0, …, p − 1} and let G be the ℚ_{σ}-group generated by the same elements. The series of groups that provide the required examples are the factor-groups G_{n} = G/G^{p^{n}}, n ∈ ℕ. Being of exponent p^{n}, nilpotent, and finitely generated (also as an abstract group), G_{n} is a finite p-group for any n. Let C_{n} = {x ∈ G | φ(x) ∈ xG^{p^{n}}} be the full preimage of C_{G_{n}}(φ).

Let p^{m_{i}} denote the highest power of p dividing n_{i} = n_{i}(c), where the n_{i} are integers from lemmas of Section 1.

Lemma 8. For any n ≥ m_{8} + 1 we have C_{n} ⊆ p^{n−m_{8}−1}M.

Proof. If x ∈ C_{n}, then φ^{j}(x) ∈ xG^{p^{n}} ⊆ x + p^{n−m_{8}}M by Lemma 6. Since C_{N}(φ) = 0, we have x + φ(x) + φ^{2}(x) + ⋯ + φ^{p−1}(x) = 0. Hence px ∈ p^{n−m_{8}}M. This implies that x ∈ p^{n−m_{8}−1}M since the additive group of M has no torsion. □

Lemma 9. For every n the order |C_{G_{n}}(φ)| is p-bounded (by a positive integer independent of n but depending on p only).

Proof. Let T be a transversal to G^{p^{n}} in C_{n}, so that |T | = |C_{G_{n}}(φ)|. We define the mapping α : T → p^{n−m_{8}−1}M/p^{n+m_{7}}M by the rule α(t) = t+p^{n+m_{7}}M. The image is indeed in p^{n−m_{8}−1}M/p^{n+m_{7}}M by Lemma 8. This mapping is injective by Lemma 6. Hence |T | ≤ |p^{n−m_{8}−1}M/p^{n+m_{7}}M|. The additive group of p^{n−m_{8}−1}M/p^{n+m_{7}}M has exponent p^{m_{7}+m_{8}+1}. As a ℚ_{σ}-module p^{n−m_{8}−1}M/p^{n+m_{7}}M is generated by p-boundedly many elements, namely, the images of the p^{n−m_{8}−1}th multiples of basic commutators of weight ≤ h in the elements {n_{i}^{φ^{j}} | i = 1, …, h; j = 0, …, p−1}. Hence, being a p-group, p^{n−m_{8}−1}M/p^{n+m_{7}}M is also generated by the same set as an abstract group. As a result, this group has p-bounded order, and hence so does C_{G_{n}}(φ). (Here we could assume n to be large enough with respect to any function of p, since otherwise even the order of G/G^{p^{n}} is p-bounded.) □

Lemma 10. For every a ≥ 0 there exists n ≥ a such that the nilpotency class of G^{p^{a}}/G^{p^{n}} is exactly h.

Proof. Given a, we simply show that [n_{1}^{p^{a}}, …, n_{h}^{p^{a}}]_{G} ∉ G^{p^{n}} for some n. Since G^{p^{n}} ⊆ p^{n−m_{4}}M by Lemma 2, it suffices to show that [n_{1}^{p^{a}}, …, n_{h}^{p^{a}}]_{G} ∉ p^{n−m_{4}}M. Since the nilpotency class is h, the group commutator of weight h is equal to the same Lie ring commutator:

[n_{1}^{p^{a}}, …, n_{h}^{p^{a}}]_{G} = [p^{a}n_{1}, …, p^{a}n_{h}]_{L} = p^{ah}[n_{1}, …, n_{h}]_{L}.

The additive group of M is finitely generated; hence ⋂_{i} p^{i}M = 0. Since [n_{1}, …, n_{h}]_{L} ≠ 0, there is j such that [n_{1}, …, n_{h}]_{L} ∉ p^{j}M. Then p^{ah}[n_{1}, …, n_{h}]_{L} ∉ p^{j+ah}M since M is torsion-free. Hence the lemma is true for n ≥ j + ah + m_{4}. □

By Lemmas 9 and 10 the groups G_{n} provide the examples proving the following.

Theorem 5. There is no function g(p, m) such that for every finite p-group G admitting an automorphism of order p with exactly p^{m} fixed points the nilpotency class of G^{p^{g(p,m)}} is ≤ h_{ℚ}(p) − 1. □

Now we turn to Khukhro’s theorem [7] on finite p-groups with an automorphism of order p^{n} with p^{m} fixed points. Again, in [7] it was Kreknin’s function k = k_{ℤ} defined via a graded Lie ℤ-ring, but quite a similar argument as for Theorem 4 allows one to replace k_{ℤ}(p^{n}) by the possibly smaller value k_{ℚ}(p^{n}) defined via graded Lie ℚ-algebras.

Theorem 6. If a finite p-group P admits an automorphism of order p^{n} with exactly p^{m} fixed points, then P has a subgroup of (p, m, n)-bounded index which has derived length ≤ 2k_{ℚ}(p^{n}). □

It is yet unclear whether one can achieve the bound k(p^{n}) for the derived length. We shall now show that here one cannot make the bound for the derived length better than k_{ℚ}(p^{n}). There remains a gap between k_{ℚ}(p^{n}) in the examples and 2k_{ℚ}(p^{n}) in Theorem 6.

The corresponding examples can be constructed in essentially the same way as in the proof of Theorem 5, provided we could start with a nilpotent Lie ℚ-algebra of derived length exactly k = k_{ℚ}(p^{n}) that admits a fixed-point-free automorphism of order p^{n}. We now show that such an algebra does really exist.

Let L be a free (ℤ/p^{n}ℤ)-graded Lie ℚ-algebra on countably many free generators. One can think of L as an abstract free Lie ℚ-algebra on free generators x_{ij}, i = 0, 1, …, p^{n}−1, j = 1, 2, …, where the first index is used to indicate the component of the grading. Then the (ℤ/p^{n}ℤ)-grading is given by the decomposition L = L_{0} ⊕ ⋯ ⊕ L_{p^{n}−1}, where L_{k} is the span of all commutators in the x_{ij} whose sum of the first indices is k (mod p^{n}). Of course,

L is homogeneous with respect to the free generators x_{ij}, that is, L = ⨁_{i=1}^{∞} W_{i}, where W_{t} is

the span of all commutators of weight t in the x_{ij}.

Now let I be the ideal of L generated by L_{0}. By Kreknin’s theorem L/I is soluble of p^{n}-bounded derived length k = k_{ℚ}(p^{n}), which defines Kreknin’s function.

The ideal I is also homogeneous: I = ⨁_{i}(I ∩ W_{i}). Indeed, it is spanned by commutators of the form [l_{0}, y_{1}, …, y_{s}], where l_{0} ∈ L_{0} and y_{i} are arbitrary elements from L. But L_{0} is spanned by certain commutators in the x_{ij} as well as L is. Hence I is spanned by certain commutators in the x_{ij} and therefore is homogeneous.

Now L/I = ⨁_{i≥1}(W_{i}/(I ∩ W_{i})), and γ_{m}(L/I) = ⨁_{i≥m}(W_{i}/(I ∩ W_{i})) for any m. Let δ_{n}(y_{1}, …, y_{2^{n}}) = 0 be the identity of solubility of derived length n. Since the derived length of L/I is k, there are some z_{i} among the images of the x_{ij} such that δ_{k−1}(z_{1}, …, z_{2^{k−1}}) ≠ 0 in L/I. Since δ_{k−1}(z_{1}, …, z_{2^{k−1}}) belongs to the image of W_{2^{k−1}}, the image of δ_{k−1}(z_{1}, …, z_{2^{k−1}}) is non-trivial also in L/(I + γ_{2^{k−1}+1}(L)). The latter Lie algebra is nilpotent of class 2^{k−1}. It has a fixed-point-free automorphism of order p^{n} acting on the image of L_{i} by multiplication by ω^{i}, where ω is a primitive p^{n}th root of unity.

Now the same construction as in the proof of Theorem 5 can be used to prove the following result.

Theorem 7. There is no function g(p, m, n) such that if a finite p-group G admits an automorphism of order p^{n} with exactly p^{m} fixed points, then the derived length of G^{p^{g(p,m,n)}} is ≤ k_{ℚ}(p^{n}) − 1. □

One can construct similar examples, but based on another prime q ≠ p, that is, groups G/G^{q^{n}} as above with σ being the set of all primes other than q. As a result, we have the following result.

Theorem 8. For any two distinct primes p and q there exist finite q-groups of nilpotency class exactly h_{ℚ}(p) admitting a fixed-point-free automorphism of order p. □

In Higman’s paper [4] such examples were constructed (also using the Baker–Hausdorff formula) only for sufficiently large primes q. As we noted above, in the coprime situation we have a hypothetical gap between h_{ℚ}(p), the nilpotency class in the example, and h_{ℤ}(p), the upper bound for the class.

Using the same method we can also construct examples of finite q-groups of derived length exactly k_{ℚ}(n) admitting a fixed-point-free automorphism of coprime order n. But so far bounding the derived length of such groups, by any function of n, is still an open problem.

References

1. J. L. Alperin, Automorphisms of solvable groups, Proc. Amer. Math. Soc., 13 (1962), 175–180.

2. N. Blackburn, Conjugacy in nilpotent groups, Proc. Amer. Math. Soc., 16 (1965), 143–148.

3. N. Bourbaki, Lie groups and algebras, Chapters 1–3, Springer, Berlin et al., 1980.

4. G. Higman, Groups and rings which have automorphisms without non-trivial fixed elements, J. London Math. Soc. (2), 32 (1957), 321–334.

5. I. Hughes, Groups with fixed-point-free automorphisms, C. R. Math. Rep. Acad. Sci. Canada, 7 (1985), 61–66.

6. E. I. Khukhro, Finite p- groups admitting an automorphism of order p with a small number of fixed points, Mat. Zametki, 38 (1985), 652–657 (Russian); English transl., Math. Notes., 38 (1986), 867–870.

7. E. I. Khukhro, Finite p- groups admitting p-automorphisms with few fixed points, Mat. Sbornik, 184 (1993), 53–64; English transl., Russian Acad. Sci. Sbornik Math., 80 (1995), 435–444.

8. E. I. Khukhro, p-Automorphisms of finite p-groups (London Math. Soc. Lecture Note Ser. 246), Cambridge Univ. Press, 1998.

9. V. A. Kreknin, The solubility of Lie algebras with regular automorphisms of finite period, Dokl. Akad. Nauk SSSR, 150 (1963), 467–469 (Russian); English transl., Math. USSR Doklady, 4 (1963), 683–685.

10. V. A. Kreknin, A. I. Kostrikin, Lie algebras with regular automorphisms, Dokl. Akad. Nauk SSSR, 149 (1963), 249– 251 (Russian); English transl., Math. USSR Doklady, 4, 355–358.

11. M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Supr., 71 (1954), 101–190.

12. N. Yu. Makarenko, Almost regular automorphisms of prime order, Sibirsk. Mat. Zh., 33 (1992), No 5, 206–208 (Russian). English transl., Siberian Math. J., 33, 932–934.

13. A. I. Mal’cev, Nilpotent groups without torsion, Izv. Akad. Nauk SSSR, Ser. Mat., 13 (1949), 201–212 (Russian).

14. Yu. Medvedev, p-Groups, Lie p-rings and p-automorphisms, J. London Math. Soc., 58, No 1 (1998), 27–37.

15. Yu. Medvedev, p-Divided Lie rings and p-groups, Preprint, Univ. of Wisconsin, Madison, 1994.

16. A. Shalev, On almost fixed point free automorphisms, J. Algebra, 157 (1993), 271– 282.

17. T. Weigel, Exp and Log functors for the categories of powerful p-central groups and Lie algebras, Habilitationsschrift, Freiburg i. Br., 1994.