Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Let $f(x)$ be a non-zero polynomial with integer coefficients. An automorphism $\varphi$ of a group $G$ is said to satisfy the elementary abelian identity $f(x)$ if the linear transformation induced by $\varphi$ on every characteristic elementary abelian section $S$ of $G$ is annihilated by $f(x)$. We prove that if a finite (soluble) group $G$ admits a fixed-point-free automorphism $\varphi$ satisfying an elementary abelian identity $f(x)$, where $f(x)$ is a primitive polynomial, then the Fitting height of $G$ is bounded in terms of $\operatorname{deg}(f(x))$. We also prove that if $f(x)$ is any non-zero polynomial and $G$ is a $\sigma'$-group for a finite set of primes $\sigma=\sigma(f(x))$ depending only on $f(x)$, then the Fitting height of $G$ is bounded in terms of the number $\operatorname{irr}(f(x))$ of irreducible factors in the decomposition of $f(x)$. These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length $\alpha (|\varphi|)$ of $\langle\varphi\rangle$ when $\operatorname{deg} (f(x))$ or $\operatorname{irr}(f(x))$ is small in comparison with $\alpha (|\varphi|)$.


Introduction
An automorphism ϕ of a group G is said to be fixed-point-free if C G (ϕ) = 1, that is, the only fixed point of ϕ is 1. By the celebrated theorem of Thompson [33], a finite group with a fixed-point-free automorphism of prime order p is nilpotent (and the nilpotency class is bounded in terms of p by Higman's theorem [10], with the bound made effective by Kreknin and Kostrikin [24,25]). Based on the classification of finite simple groups, Rowley [31] proved the solubility of a finite group admitting a fixed-point-free automorphism of any order (not necessarily coprime to |G|).
If a finite (soluble) group G admits a fixed-point-free automorphism ϕ of coprime order, then the Fitting height of G is at most the composition length α(|ϕ|) of ϕ , which is the best possible bound. In this most general form, this is a special case of Berger's theorem [3]. Earlier results under certain restrictions on the primes dividing |ϕ| were obtained by Shult [32] and Gross [8]. Their papers contained important so-called nonmodular Hall-Higman type theorems (with Gross also referring to Dade's seminar notes of 1964).
A special case of Dade's theorem [4] gives a bound for the Fitting height of G admitting a fixed-point-free automorphism ϕ of non-coprime order. The bound furnished by Dade's theorem is exponential in α(|ϕ|); a significant improvement to a quadratic bound was recently obtained by Jabara [17].
There are also many results, starting from the papers of Thompson [34] and Dade [4], on bounding the Fitting height of finite soluble groups in terms of fixed points and orders of their (not necessarily cyclic) groups of automorphisms. We refer the reader to the survey of Turull [35], who obtained some of the best results in this area.
In this paper we consider finite groups G admitting a fixed-point-free automorphism ϕ that satisfies an additional polynomial identity f (x) for an arbitrary polynomial f (x) ∈ Z [x]; see the precise definition below. Imposing this additional condition on the automorphism is justified by obtaining upper bounds for the Fitting height of G independent of α(|ϕ|). Instead, we obtain bounds for the Fitting height of G depending only on the degree of f (x) (Theorem 1.4) or on the number irr(f (x)) of different irreducible factors in the decomposition of f (x) (Theorem 1.5). The bounds obtained in our theorems are stronger than the known bound α(|ϕ|) for coprime automorphisms (or a bound in terms of α(|ϕ|) in general) when deg f (x) or irr(f (x)) is small in comparison with α(|ϕ|). Our result in terms of deg(f (x)) applies to any finite group G (with a natural and unavoidable condition on the polynomial f (x)). The result in terms of irr(f (x)) applies to all finite σ ′ -groups, where σ = σ(f (x)) is a finite set of primes depending only on f (x).
Earlier this approach to the study of finite groups with fixed-point-free automorphisms was proposed by the second author [27,28,29].
We now pass to precise definitions and statements of the results. Let ϕ be an automorphism of a group G. Definition 1.1. We say that a polynomial f (x) = a 0 + a 1 x + · · · + a d x d ∈ Z[x] is an ordered identity of ϕ if When G is an abelian group, the order of the factors here is unimportant, and then we can simply say that f (x) is an identity of ϕ.
is an identity of the automorphisms induced by ϕ on every characteristic elementary abelian section of G. In this case, we also say that ϕ satisfies the elementary abelian identity f (x).
It is clear that an ordered identity of ϕ is also an elementary abelian identity of ϕ, but the converse is not true in general.
is an elementary abelian identity of ϕ ∈ Aut G and S is an elementary abelian p-group that is a characteristic section of G, then the F p -linear transformation induced by ϕ on S regarded as a vector space over F p is annihilated by the polynomial f (x) (reduced modulo p) in the ordinary sense of linear algebra.
We further recall that a polynomial f ( Examples show that if we drop the condition that f (x) is primitive, then one cannot obtain a bound for the Fitting height of G in terms of deg(f (x)). We conjecture that such a bound could possibly be obtained for an arbitrary polynomial f (x) = 0 if "elementary abelian identity" is replaced with the stronger "ordered identity".
A stronger bound for the Fitting height is obtained for an arbitrary polynomial f (x) under an additional restriction on the prime divisors of the order of the group. Recall that if σ is a set of primes, then a finite group G is a σ ′ -group if |G| is not divisible by any prime in σ. Let irr(f (x)) denote the number of different irreducible factors of f (x) in Z[x] (counted without multiplicities). Theorem 1.5. Suppose that a finite (soluble) group G admits a fixed-point-free automorphism satisfying an elementary abelian identity f (x) ∈ Z[x], where f (x) is a non-zero polynomial. There is a finite set of primes σ = σ(f (x)) depending only on f (x) such that if G is a σ ′ -group, then the Fitting height of G is at most 2 + irr(f (x)) 2 .
The set of primes σ(f (x)) in this theorem is described explicitly in Definition 5.2. Earlier the second author [28] proved that, for ordered identities f (x) with irreducible polynomial f (x), Theorem 1.5 holds with a sharp bound 1 for the Fitting height (for a different finite set of primes σ(f (x))). In [29] the second author proved a result similar to Theorem 1.5 bounding the Fitting height of G by the number of irreducible factors of f (x) counting multiplicities for a certain class of polynomials f (x). The bound for the Fitting height that we obtain in Theorem 1.5 may not be best-possible, but importantly it only depends on the number of different irreducible factors irr(f (x)) and the theorem holds for any non-zero polynomial f (x). The bound for the Fitting height in Theorem 1.4 is weaker, in terms of deg(f (x)), but the advantage is that it does not impose any restrictions on the prime divisors of |G| provided that the polynomial f (x) is primitive (for example, when f (x) is monic).
Let G be a finite (soluble) group admitting a fixed-point-free automorphism ϕ satisfying an elementary abelian identity f (x) ∈ Z[x]. The first step in the proof of both Theorems 1.4 and 1.5 is to use Hall-Higman type theorems (with certain modifications) for obtaining a reduction to the situation where ϕ has order bounded in terms of deg(f (x)). Then the proof of Theorem 1.4 follows by an application of a special case of Dade's theorem [4], or rather Jabara's [17] recent improvement for the bound for the Fitting height of a finite group admitting a fixed-point-free automorphism of not necessarily coprime order.
In the proof of Theorem 1.5 it is the Shult-Gross-Berger theorem that is ultimately applied after a reduction to the case of a (possibly different) coprime automorphism such that the number α(|ϕ|) of prime factors in |ϕ| counting multiplicities is bounded in terms of irr(f (x)). This becomes possible after defining a 'forbidden' finite set of primes σ depending only on f (x) such that for σ ′ -groups G the automorphism ϕ can be assumed to be of coprime order and 'non-exceptional' in the sense of Hall-Higman type theorems, while f (x) can be assumed to be a product of cyclotomic polynomials. Then Hall-Higman type theorems are applied again to reduce to a situation where α(|ϕ|) is bounded in terms of irr(f (x)) and an application of the Shult-Gross-Berger theorem finishes the proof.

Preliminaries
Suppose that a group A acts by automorphisms on a group B. We use the usual notation for commutators [ In particular, then A/C A (B) embeds in the automorphism group Aut B. If ϕ is an automorphism of a group G and S is a ϕ-invariant section of G, then we denote by ϕ| S the automorphism induced by ϕ on S; sometimes we denote the induced automorphism by the same letter when this causes no confusion.
We recall the well-known property of coprime actions.
Lemma 2.1. Let ϕ be an automorphism of coprime order of a finite group G. If N is a normal ϕ-invariant subgroup, then C G/N (ϕ) = C G (ϕ)N/N . In particular, if ϕ acts trivially on every factor of some subnormal ϕ-invariant series of G, then ϕ = 1.
It is also known that if ϕ is a coprime automorphism of a finite group G, then for every prime q there is a ϕ-invariant Sylow q-subgroup. It is important for us that a similar property and an analogue of Lemma 2.1 hold for fixed-point-free automorphisms of any order, not necessarily coprime to the order of the group. Henceforth we freely use the fact that a finite group with a fixed-point-free automorphism is soluble [31]. Proof. We only need to replace Sylow's theorem with Hall's theorem in the proof of [7, Theorem 10.1.2] to obtain the lemma for arbitrary sets of primes σ. The following result in its most general form is a special case of Berger's theorem [3].
Earlier results under certain restrictions on the primes dividing |ϕ| were obtained by Shult [32] and Gross [8]. Recall that α(n) denotes the number of prime divisors of n counting multiplicities.

Theorem 2.4 (Shult-Gross-Berger).
If a finite (soluble) group G admits a fixed-pointfree automorphism of coprime order n, then the Fitting height of G is at most α(n).
For fixed-point-free automorphisms of non-coprime order, the first (exponential) bound for the Fitting height was obtained as a special case of a theorem by Dade [4]; the quadratic bound was recently obtained by Jabara [17, Corollary 1.2].
Theorem 2.5 (Dade-Jabara). If a finite (soluble) group G admits a fixed-point-free automorphism of order n, then the Fitting height of G is bounded in terms of α(n) and is at most 7α(n) 2 .
We collect in the next lemma well-known facts about minimal polynomials of linear transformations (which are assumed monic). If ϕ is an automorphism of a vector space V over a field F , we regard V as a right F ϕ -module. Lemma 2.6. Suppose that ϕ is an automorphism of a vector space V over a field K.
(a) If ϕ is regarded as a linear transformation of the vector space V ⊗ K K 1 obtained by extending the ground field to K 1 , then the minimal polynomial remains the same. (b) If ϕ is diagonalizable, then the degree of the minimal polynomial of ϕ is equal to the number of different eigenvalues of ϕ. (c) There is a vector v ∈ V such that the minimal polynomial of ϕ for v, that is, the polynomial g(x) of smallest degree such that vg(ϕ) = 0, is the same as the minimal polynomial of ϕ for the whole space V . (d) The degree of the minimal polynomial of ϕ is equal to the maximum dimension of the subspace spanned by an orbit of a vector under the action of ϕ .
The degree of the minimal polynomial of any power ϕ k is at most the degree of the minimal polynomial of ϕ.
Proof. All these properties are well-known, but we still indicate some references. Recalling Definition 1.1 we can say that a linear transformation ϕ of a vector space satisfies an identity f (x) ∈ Z[x] if ϕ is annihilated by f (x) (reduced in the ground field).
Lemma 2.7. Let ϕ be an automorphism of finite order |ϕ| of a vector space V over a field of characteristic q coprime to |ϕ|. Suppose that ϕ satisfies the identity are some cyclotomic polynomials such that gcd(q, n i ) = 1.
(a) For any s ∈ N the power ϕ s satisfies the identity Proof. Let V be the vector space obtained from V by extension of the ground field to an algebraically closed field K, so that ϕ naturally becomes a K-linear transformation of V .
Since the order of ϕ is coprime to the characteristic, the transformation ϕ of finite order is diagonalizable over K. The eigenvalues of ϕ are roots of f (x) (regarded as a polynomial in K[x]).
(a) The eigenvalues of ϕ s are the s-th powers of the roots of ϕ, and therefore they are If |ϕ| is divisible by a power of a prime p k , then at least one of the eigenvalues of ϕ has multiplicative order divisible by p k and therefore can be a root of a cyclotomic polynomial Φ n i (x) only if n i is divisible by p k . Proof. Let f (x), g(x) ∈ Z[x] be elementary abelian identities of ϕ, and let h(x) ∈ Z[x]. Let S be a characteristic elementary abelian section of G regarded as a right Z ϕ -module. Then by definition x(f (ϕ)+g(ϕ)) = 0+0 = 0 and x(f (ϕ)h(ϕ)) = 0h(ϕ) = 0 for all x ∈ S. Thus, both f (x) + g(x) and f (x)h(x) are elementary abelian identities of ϕ.

Remark 2.9.
In what follows, we will consider polynomials in the ring Z[x] and also polynomials with coefficients in a field of prime characteristic. Unless specifically stated otherwise, every polynomial is assumed to belong to Z[x]. Moreover: divisors, greatest common divisors, decompositions into irreducible factors, etc. are to be understood in the ring Z[x], unless stated otherwise.
The prime number theorem, commonly attributed to Hadamard and de la Vallée Poussin, establishes that the prime counting function π(x) satisfies π(x) ∼ x/ ln(x), that is, lim x→∞ π(x)/(x/ ln(x)) = 1. We will use the following estimate due to Rosser and Schoenfeld [30, Corollary 1] that is valid for all x > 1.

Hall-Higman type theorems
In this section we lay out the foundations of further proofs by producing Hall-Higman type theorems, both of 'modular' and 'non-modular' kind. First we state the celebrated Hall-Higman Theorem B, which is of 'modular' kind, dealing with the minimal polynomial of an element of order p m in a linear group over a field of characteristic p. . Let H be a p-soluble linear group over a field of characteristic p, with no normal p-subgroup greater than 1. If g is an element of order p m in H, then the minimal polynomial of g is (x − 1) d = 0, where d = p m , unless there is an integer m 0 m such that p m 0 − 1 is a power of a prime q for which a Sylow q-subgroup of H is non-abelian, in which case, if m 0 is the least such integer, Note that the degree d of the minimal polynomial of g satisfies d p m − p m−1 in all cases.
We now consider Hall-Higman type results in so-called 'non-modular' situations, which analyse the minimal polynomial of an element of order p m in a linear group over a field of characteristic not equal to p.
Suppose that L is an extra-special r-group of order r 2t+1 , and η is a cyclic group of order p k for a prime p = r acting (not necessarily faithfully ) by automorphisms on L such that the induced automorphismη of L has order p m , acts regularly on the set L/Z(L)\{1} with all orbits of length p m , and centralizes Z(L). Suppose that the semidirect product L η acts by linear transformations on a vector space V over an algebraically closed field K whose characteristic does not divide p·r, and suppose that V is an irreducible KL η -module and a faithful and homogeneous KL-module.
(a) Then η as a linear transformation of V has at least p m − 1 different eigenvalues, so that the minimal polynomial of η on V has degree at least p m − 1. Part (a) does not seem to have appeared in the literature; its proof is similar to the well-known proof for (b), which was really using only the number of eigenvalues of η. We write this modified proof in full for the benefit of the reader.
By [14,Lemma IX.2.5], V is an irreducible KL-module and has dimension dim K V = r t as a vector space over K. Then the enveloping algebra E for L coincides with the full matrix algebra [7, Theorem 3.6.2] and dim K E = r 2t . The elements of Z(L) are represented by scalar transformations and multiplication by such a transformation in E is equivalent to multiplication by the corresponding field element. Therefore any set of representatives of the r 2t cosets of Z(L) forms a basis of the algebra E.
The element η naturally acts on E and we calculate the dimension of the centralizer of η in E. Since η p m belongs to the centre of the semidirect product L η by hypothesis, the subgroup η p m is represented by scalar transformations and the action of η on E factors through to the action of η = η / η p m . Sinceη acts regularly on L/Z(L) \ {1} with orbits of length p m , it acts regularly on the elements of some basis of E except for one element of this basis that corresponds to Z(L) and belongs to C E (η). In every subspace of E spanned by a non-trivial orbit of η on this basis, the fixed point subspace is one-dimensional (spanned by the sum of the elements of the orbit). It follows that dim C E (η) = r 2t − 1 p m + 1.
We now calculate the same quantity in another way. Let a i , i = 1, . . . , l, be the multiplicities of the l distinct eigenvalues of the linear transformation η. Then the matrix of η in some basis is block-diagonal consisting of l scalar blocks with different eigenvalues on the diagonals. The centralizer of this matrix in the full matrix algebra consists of all block-diagonal matrices with the same block-partition. Therefore, If l = p m , there is nothing to prove. Suppose that l p m − 1.
and as a result, This is equivalent to saying that r t + 1 p m . Using this inequality we now obtain that Therefore, all the inequalities are actually equalities, in particular, l = p m − 1. Thus, l p m − 1 in all cases, as required.
The following lemma readily follows from Lemma 3. Lemma 3.3. Suppose that r is a prime, R is an r-group, and ψ is a cyclic group of order p k for a prime p = r acting (not necessarily faithfully ) by automorphisms on R such that the induced automorphism of R has order p m . Suppose that the semidirect product R ψ acts by linear transformations on a vector space V over a field K whose characteristic does not divide p · r, and suppose that V is a faithful KR-module.
(a) Then the minimal polynomial of ψ on V has degree at least p m − p m−1 . (b) If in addition k = m that is, ψ acts faithfully on R, then either the minimal polynomial of ψ on V is x p k − 1, or there exist positive integers k 0 k and t such that r t = p k 0 − 1 and if p = 2, then t = 1.
Note that in the case of r t = p k 0 − 1 either r = 2 or p = 2 and r is a Mersenne prime.
Proof. (a) We can assume that the field K is algebraically closed, since the hypotheses of the lemma remain valid for R ψ regarded as a group of linear transformations of the vector space obtained by extending the ground field, while the minimal polynomial of ψ will be the same by Lemma 2.6(a). Since the characteristic of the field K is coprime to p · r, the KM ψ -module V is completely reducible. Let W be an irreducible KM ψ -submodule of V on which the subgroup M = [M, ψ p m−1 ] acts non-trivially. Applying Clifford's theorem [7, Theorem 3.4.1] with respect to the normal subgroup M we decompose W = W 1 ⊕ · · · ⊕ W p s into a sum of homogeneous KM-submodules, which are transitively permuted by ψ with ψ p s being the stabilizer of W 1 in ψ . Then ψ p s is also the stabilizer of every W i since ψ is abelian. Note that p s p m . Let L be the image of M in its action on W 1 . Then W 1 is an irreducible KL ψ p s -module and a homogeneous KL-module.
If L is abelian, then it is cyclic and central, so that [L, ψ p s ] = 1 and therefore [L ψ i , ψ p s ] = 1 for all i and [L, ψ p s ] acts trivially on W , whence p s = p m . Then ψ has minimal polynomial on V of degree at least p m by Lemma 2.6(e).
If L is not abelian, then it is an extra-special r-group, ψ p s acts trivially on Z(L) and has regular orbits of length |ψ p s |/|ψ p m | = p m−s on L/Z(L). Thus the semidirect product L ψ p s and the KL ψ p s -module W 1 satisfy the hypotheses of , but for completeness we also derive this result here from Lemma 3.2 (b). Assuming k = m (that is, ψ acts faithfully on R), we repeat the above arguments. When L is abelian, we obtain that ψ has minimal polynomial on V of degree at least p k by Lemma 2.6(e), so it must be x p k − 1. When L is extraspecial, the semidirect product L ψ p s and the KL ψ p s -module W 1 satisfy the hypotheses of Lemma 3.2(b), by which either the minimal polynomial of ψ p s has degree at least p m−s or r t = p k−s − 1 and if p = 2, then t = 1. In the first case, by Lemma 2.6(e) the minimal polynomial of ψ has degree at least p s p k−s = p k and then it must be x p k − 1. The second case corresponds to the other alternative in part (b).
Of course, a crude bound for α(|ϕ|Ḡ /F (Ḡ) |) immediately follows from a bound for |ϕ|Ḡ /F (Ḡ) |, but we included a sharper bound, which is easily obtained in the proof.
Note that the condition that the polynomial f (x) does not vanish modulo prime divisors of |G| is automatically satisfied in Theorem 1.4. Moreover, the definition of the set σ(f (x)) will ensure that this condition is also satisfied in Theorem 1.5. In what follows, when we will be using the fact that ϕ satisfies f (x) on some elementary abelian characteristic section, it will only matter that f (x) reduced modulo some prime is non-zero, so the minimal polynomial of ϕ regarded as a linear transformation of this section has degree at most deg(f (x)).
Letφ be the induced automorphism of VḠ, which is fixed-point-free by Lemma 2.3. Let p be a prime divisor of |φ| and let |φ| = s p p kp for s p coprime to p. For brevity let us write simply s = s p and k = k p when we focus on this prime p. Note that it may happen that p = q or p = q.
We write ψ =φ s , which is a generator of the Sylow p-subgroup of φ . Let p m be the order of the automorphism induced by ψ on aφ-invariant Hall p ′ -subgroup H ofḠ, which exists by Lemma 2.2. Proof. We claim that ψ p m ∈ O p (Ḡ φ ) F (Ḡ φ ). Indeed, by definition ψ p m centralizes a ϕ-invariant Hall p ′ -subgroup H ofḠ, as well as, obviously, the Hall p ′ -subgroup of φ . Hence ψ p m centralizes a Hall p ′ -subgroup H 1 ofḠ φ . SinceḠ φ = H 1 P , where P is a Sylow p-subgroup containing ψ p m , all conjugates of ψ p m belong to P and therefore generate a normal p-subgroup contained in O p (Ḡ φ ). Thus, ψ p m ∈ F (Ḡ φ ), and since F (Ḡ) =Ḡ ∩ F (Ḡ φ ), the order of the automorphism ofḠ/F (Ḡ) induced by ψ is at most p m . Proof. By Lemma 2.2 for every prime divisor r of |H| there is aφ-invariant Sylow rsubgroup of H. Clearly, ψ must act as an automorphism of order p m on at least one such Sylow subgroup. Let R be a ψ-invariant Sylow r-subgroup ofḠ for some prime r = p on which ψ acts with order p m , that is, on which ψ p m−1 acts non-trivially. The semidirect product R ψ acts by linear transformations on V , and V is a faithful F q R-module. We claim that p m − p m−1 deg(f (x)), which implies the required bound.
First suppose that r = q. Then the action of the semidirect product R ψ on V gives rise to a Hall-Higman type situation, which may be 'modular' when p = q, or 'non-modular' when p = q.
In the 'non-modular' case p = q we apply Lemma 3.3(a), by which the minimal polynomial of ψ on V has degree at least p m − p m−1 . By Lemma 2.6(f) the minimal polynomial of ϕ on V must also have degree at least p m −p m−1 . Since ϕ satisfies f (x) on V , we obtain that p m − p m−1 deg(f (x)).
In the 'modular' case p = q, we apply Theorem 3.1 after certain preparation. Sincē G = G/O p ′ ,p (G) and V is the Frattini quotient of O p ′ ,p (G)/O p ′ (G), while ψ is a p-group, we have O p ′ (VḠ ψ ) = 1. Since ψ / ψ p m acts faithfully on a Hall p ′ -subgroup ofḠ, while ψ p m ∈ O p (Ḡ ψ ) as shown in the proof of Lemma 4.2, we further obtain Let U be the Frattini quotient of O p ′ ,p (VḠ ψ )/O p ′ (VḠ ψ ), on which the group VḠ ψ /O p ′ ,p (VḠ ψ ) acts faithfully by conjugation satisfying the hypotheses of Theorem 3.1. Since the order of the automorphism of U induced by ψ is p m , by Theorem 3.1 the minimal polynomial of ψ on U has degree at least p m − p m−1 . We have Since both factors of U are ψ-invariant and ψ p m / ψ p m+1 is centralized by ψ, the minimal polynomial of ψ on the first factor, and therefore on V , must also have degree at least p m − p m−1 . By Lemma 2.6(f) the minimal polynomial ofφ on V must have degree at least p m − p m−1 . Sinceφ satisfies f (x) on the characteristic abelian section V , we obtain that p m − p m−1 deg(f (x)). Now suppose that r = q. In this case R is a q-group, and we consider the action of the semidirect product R ψ on F (Ḡ), which is a nilpotent q ′ -group containing its centralizer. Since R acts faithfully on F (Ḡ), there is some Sylow t-subgroup T of F (Ḡ) for t = r on which the subgroup [R, ψ p m−1 ] acts non-trivially. Let R = R/C R (T ). Note that by the choice of T the automorphism induced by ψ on R has order p m . Then the action of the semidirect product R ψ on the Frattini quotient U of T again gives rise to a Hall-Higman type situation, which may be 'modular' when p = t, or 'non-modular' when p = t. Using the same Hall-Higman type arguments as above, based either on for each i ∈ {1, . . . , l}, and therefore the bound |ϕ|Ḡ /F (Ḡ) | (2 deg(f (x))) 2 deg(f (x)) .

Proof of Theorem 1.5
In Theorem 1.5, for a finite soluble group G with a fixed-point-free automorphism ϕ satisfying an elementary abelian identity f (x) ∈ Z[x], we need to obtain a bound on the Fitting height of G in terms of irr(f (x)). As a first step, we use Proposition 4.1 to perform a reduction to the case where the order of ϕ is bounded in terms of deg(f (x)). Then, roughly speaking, we can pass to a fixed-point-free automorphism ϕ satisfying the elementary abelian identity gcd(f (x), x |ϕ| − 1), which is a product of cyclotomic polynomials the number of which is at most irr(f (x)). First we deal with this situation in the following proposition, which requires imposing additional conditions on the prime divisors of |G|. Then we finish the proof of Theorem 1.5 after defining a finite set of 'forbidden' prime divisors of |G| depending only on f (x).
Then the Fitting height h(G) of G is at most c 2 .
Note that the coprimeness conditions on the order of G ensure that ϕ is a coprime automorphism of G for which no exceptional situations arise in the 'non-modular' Hall-Higman type arguments for automorphisms induced by powers of ϕ.
Proof. We proceed by induction on the pairs (c, α(|ϕ|)) ordered lexicographically. We formally include the case c = 0 as the base of the induction, which means that f (x) = 1. By the definition of elementary abelian identity we then have u = 1 for every u ∈ S in every characteristic abelian section S of G, which of course means that G = 1, so that indeed h(G) = 0 0 2 . We may therefore assume that c 1.
First we show that the number of distinct primes dividing the order of ϕ can be assumed to be at most c, by way of possibly replacing ϕ with another fixed-point-free automorphism. Suppose that for some prime p the cyclotomic polynomial Φ p (x) does not occur in the factorization of f (x). Then, since the eigenvalues of ϕ on every elementary abelian characteristic section S of G are roots of f (x) and gcd(|G|, n i ) = 1 for every factor Φ n i (x) of f (x), the power ϕ p is also a fixed-point-free automorphism of S. Therefore ϕ p is a fixed-point-free automorphism of G. By Lemma 2.7(a), the automorphism ϕ p satisfies the elementary abelian identity c i=1 Φ n i /gcd(n i ,p) (x) with the same number of cyclotomic factors (or possibly fewer if repeats appeared). Clearly, the coprimeness hypotheses of Proposition 5.1 on |G| also hold with respect to ϕ p instead of ϕ. Since α(|ϕ p |) = α(|ϕ|)−1, induction completes the proof. Hence we can assume from the outset that for every prime p dividing |ϕ|, the cyclotomic polynomial Φ p (x) does occur in the factorization of f (x), and that therefore the number of distinct primes dividing |ϕ| is at most c.
For every prime q dividing |G|, considerḠ = G/O q ′ ,q (G) as above. It is sufficient to prove that h(Ḡ) c 2 − 1 for every q, as then the Fitting height of G will be at most c 2 because F (G) = q O q ′ ,q (G). We fix the prime q for what follows. The group G acts faithfully on the Frattini quotient V of O q ′ ,q (G)/O q ′ (G). Letφ be the induced automorphism of VḠ. Let p be a prime divisor of |φ| and let |φ| = s p p kp for s p coprime to p. For brevity we write s = s p and k = k p when we focus on this prime p, and let ψ =φ s be a generator of the Sylow p-subgroup of φ .
There must be some factor Φ n j (x) of f (x) with p k dividing n j , since p k obviously divides the order of ϕ. Indeed, the (cyclic) Sylow p-subgroup of ϕ must act faithfully on some characteristic elementary abelian section S of G by Lemma 2.1. Then the automorphism induced by ϕ on S has some eigenvalues of order divisible by p k and therefore at least one of the factors Φ n j (x) of f (x) must have n j divisible by p k by Lemma 2.7 (b).
If the automorphism induced by ψ onḠ/F (Ḡ) has order less than p k = |ψ|, then ϕ has no primitive roots of unity of order divisible by p k on any elementary abelian characteristic section ofḠ/F (Ḡ). Then the polynomial f (x)/Φ n j (x), with n j divisible by p k , is an elementary abelian identity of the automorphism ofḠ/F (Ḡ) that is induced by ϕ. Since f (x)/Φ n j (x) has c − 1 cyclotomic factors, by the induction hypothesis the Fitting height ofḠ/F (Ḡ) is at most (c − 1) 2 , so that the Fitting height ofḠ is at most (c − 1) 2 + 1 and therefore at most c 2 , as required, since c 1.
Therefore we can assume that the automorphism induced by ψ onḠ/F (Ḡ) has the same order p k as on VḠ. This condition means that an application of the Hall-Higman type arguments will result in the minimal polynomial of ψ being x p k − 1, as will follow from Lemma 3.3 (b) in view of the absence of exceptional situations.
Namely, choose a ψ-invariant Sylow r-subgroup R ofḠ for some prime r = p on which ψ acts with order p k , that is, on which ψ p k−1 acts non-trivially. The semidirect product R ψ acts by linear transformations on V , and V is a faithful F q R-module.
First suppose that r = q. Then the action of the semidirect product R ψ on V gives rise to a 'non-modular' Hall-Higman type situation, since p = q by the hypotheses on |G|. Since ψ acts faithfully on R, we can apply Lemma 3.3 (b), by which the minimal polynomial of ψ on V is x p k − 1 in view of the hypotheses on |G|.
Now suppose that r = q. In this case, R is a q-group, and we consider the action of the semidirect product R ψ on F (Ḡ), which is a nilpotent q ′ -group containing its centralizer. Since R acts faithfully on F (Ḡ), there is some Sylow t-subgroup T of F (Ḡ) for t = r on which the subgroup [R, ψ p k−1 ] acts non-trivially. Let R = R/C R (T ). Note that by the choice of T the automorphism induced by ψ on R has order p k . Then the action of the semidirect product R ψ on the Frattini quotient U of T again gives rise to a 'nonmodular' Hall-Higman type situation, since p = t. Applying Lemma 3.3 (b) we obtain in similar fashion that the minimal polynomial of ψ on T /Φ(T ) is x p k − 1.
Thus, in any case, there is an elementary abelian characteristic section S of VḠ on which the minimal polynomial of ψ is x p k − 1. We regard S as a vector space over a finite field, which we extend to an algebraically closed one. Since the automorphismφ is of order coprime to the characteristic, it is diagonalizable. Since the minimal polynomial of ψ on S is x p k − 1, the eigenvalues of ψ =φ s are all primitive p i -th roots of unity for all i = 0, 1, 2, . . . , k. At the same time, these eigenvalues are the s-th powers of the eigenvalues ofφ. Hence the eigenvalues ofφ include primitive roots of unity for which the highest power of p dividing their order ranges over all values p 0 , p 1 , p 2 , . . . , p k . Sincē ϕ satisfies the polynomial f (x) = c i=1 Φ n i (x), there must be different cyclotomic factors Φ n i (x) in f (x) such that the highest power of p dividing the n i ranges over all values p 0 , p 1 , p 2 , . . . , p k . Therefore the exponent k = k p of the highest power of p dividing |φ| satisfies k c − 1.
Since the number of distinct primes dividing |ϕ| is at most c by our assumption, we obtain that α(|φ|) c(c −1). By the Shult-Gross-Berger Theorem 2.4, the Fitting height of VḠ = V (G/O q ′ ,q ) is at most c(c − 1), so obviously the Fitting height of G/O q ′ ,q is also at most c(c − 1). Since this is true for every q, the Fitting height of G/F (G) is also at We first verify the coprimeness conditions. Clearly, gcd(|D|, 2) = 1, since G is a σ(f (x)) ′ -group. According to Lemma 5.3(b), the polynomial f (x) does not vanish modulo any prime divisor of |G|. Therefore we can apply Proposition 4.1 to G, ϕ, and f (x) in order to obtain the bound |ϕ| D | (2d) 2d . By Lemma 5.3(a), every prime divisor p of |D| satisfies p > (2d) 2d . Hence, gcd(p, |ϕ| D |!) = 1. By Lemma 5.3(c), we also have gcd(p, n 1 · · · n c ) = 1.
To prove that v(x) is an elementary abelian identity of ϕ| D , we first recall that f (x) is elementary abelian identity of ϕ| D . Since |ϕ| D | (2d) 2d , the polynomial u(x) = x (2d) 2d ! −1 is also an elementary abelian identity of ϕ| D . By Lemma 2.8, the elementary abelian identities of ϕ| D form an ideal of Z[x]. By Lemma 5.3 (d), the polynomial ρ · v(x) belongs to the ideal generated by f (x) and u(x); therefore ρ · v(x) is an elementary abelian identity of ϕ| D . We have gcd(|D|, ρ) = 1, since ρ is a product of primes in σ(f (x)) and G is a (σ(f (x))) ′ -group by hypothesis. In accordance with Euclid's algorithm, we have 1 ∈ ρ · Z[x] + |D| · Z[x] and therefore also v(x) ∈ ρ · v(x) · Z[x] + |D| · v(x) · Z[x]. Clearly, |D| is an elementary abelian identity of ϕ| D . As a result, v(x) is in the ideal of elementary abelian identities of ϕ| D , and the case v(x) = 1 is complete as explained above.
Finally, we consider the case v(x) = 1. Suppose that, for some prime divisor q of |G|, the quotient D := (G/O q ′ ,q (G))/(F (G/O q ′ ,q (G))) is non-trivial. Then we again have the bound |ϕ| D | (2d) 2d by Proposition 4.1. The polynomial u(x) is therefore again an elementary abelian identity of ϕ| D , so that v(x) is again an elementary abelian identity of ϕ| D . Since v(x) = 1, the group D is therefore trivial. This contradiction shows that, for every prime divisor q of |G|, we have h(D) = 0 and therefore h(G/O q ′ ,q (G)) 1. Since F (G) = q O q ′ ,q (G), we obtain the required bound h(G) 2 2 + irr(f (x)) 2 on the Fitting height of G.