The structure of finite groups with a f.p.f. automorphism satisfying an identity∗† Wolfgang Alexander Moens Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria E-Mail: [email protected] Tel: +43-1-4277-50468 September 10, 2020 Abstract We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. d We show, in particular, that for every polynomial r(t)= a0 + a1 · t + ··· + ad · t ∈ Z[t] that is irreducible over Q, there exist (explicit) invariants a,b,c ∈ N with the following property. Consider a finite group with a fix-point-free automorphism α : G −→ G and a0 a1 2 a2 d ad suppose that for all x ∈ G we have the equality x · α(x ) · α (x ) ··· α (x )=1G. Then G is solvable and of the form A · (B ⋊ (C × D)), where A is an a-group, B is a d b-group, C is a nilpotent c-group, and D is a nilpotent group of class at most d2 . Here, a group H is said to be an a-group (resp. b-group or c-group) if the order of every h ∈ H divides some natural power of a (resp. b or c). Contents 1 Introduction 2 1.1 Motivation ................................... 2 1.2 Mainresults................................... 4 arXiv:1810.04965v4 [math.GR] 9 Sep 2020 1.2.1 Invariants of good identities . 4 1.2.2 Structure ................................ 4 1.2.3 Existence ................................ 5 1.3 Outline ..................................... 5 2 Existence 5 2.1 Preliminaries . 5 2.2 Proof of proposition 1.2.9 ........................... 6 2.3 Proof of proposition 1.2.10 ........................... 7 ∗MSC2010: 20D45 (automorphisms of groups), 20D15 (nilpotent groups), 17B70 (graded Lie algebras). †This work was supported by the Austrian Science Fund (FWF) grants: J − 3371 − N25 (“Represen- tations and gradings of solvable Lie algebras”) and P 30842 − N35 (“Infinitesimal Lie rings: gradings and obstructions”). 1 3 The invariants 10 3.1 Preliminaries . 10 3.2 The invariant Cong(r(t))............................ 10 3.3 The invariants Discr (r(t)) and Prod(r(t)).................. 11 3.4 Arithmetically-free sets∗ of roots . 13 4 Examples 13 4.1 r(t)= t3 − 2t − 1................................ 13 4.2 Φn(t) and Ψn(t)................................ 15 5 Structure theorems for Lie rings 18 5.1 Constructing an embedding of Lie rings . 18 5.2 Bounded nilpotency of graded Lie rings . 21 5.3 Proof of theorem 1.2.6 ............................. 23 6 Structure theorems for groups 23 6.1 Proof of theorem 1.2.4 ............................. 23 6.1.1 Solvable case . 23 6.1.2 Generalcase .............................. 26 6.2 Proof of theorem 1.2.5 ............................. 26 6.3 Proof of corollary 1.2.8 ............................. 27 6.4 Proof of corollary 1.1.6 ............................. 27 7 Applications 27 7.1 Linear identities . 27 7.2 Cyclotomic identities . 28 7.2.1 Proof of theorems 7.2.5 and 7.2.6 ................... 29 7.2.2 Proof of corollaries 7.2.7 and 7.2.8 .................. 30 7.3 Anosovidentities ................................ 31 8 Closing remarks 31 1 Introduction 1.1 Motivation In this text, we study the structure of finite groups with a fix-point-free automorphism satisfying an “identity.” Before clarifying what we mean by “identity,” we recall three classic results on finite groups with a fix-point-free automorphism. Theorem 1.1.1 (Rowley [36]). If a finite group G admits a fix-point-free automorphism, then G is solvable. We recall that an automorphism is called fix-point-free (or regular) if it fixes only the trivial element of the group. Theorem 1.1.1 has a long history, going back to at least Gorenstein—Herstein [15], and it was finally confirmed by means of the classification of the finite simple groups. We refer to Rowley’s paper for a particularly short proof. By considering a special case, we can hope to obtain a stronger conclusion. Theorem 1.1.2 (J. Thompson [41]). If a finite group G admits a fix-point-free automor- phism of prime order, then G is nilpotent. Such automorphisms naturally appear in the study of groups acting simply-transitively on finite sets, and theorem 1.1.2 gives a positive answer to (what is generally known as) the Frobenius conjecture. Thompson’s proof used the celebrated p-complement theorem [42] 2 and an earlier result of Witt and Higman, but it did not require the classification of the finite simple groups. A follow-up result is: Theorem 1.1.3 (Higman [18]; Kreknin—Kostrikin [28,29]). If a nilpotent group G admits (p−1) a fix-point-free automorphism of prime order p, then its class satisfies c(G) ≤ (p−1)2 . In [18], Higman proved that there exists some upper bound for c(G) that depends only (p−1) on p. Kreknin and Kostrikin later proved in [28,29] an upper bound of (p − 1)2 . But it is conjectured that the minimal upper bound h(p) on c(G) satisfies h(p)= ⌈(p2 − 1)/4⌉. This problem has been referred to as the Higman conjecture, and it still open for primes p ≥ 11. We next make some elementary observations. Remark 1.1.4. We suppose, as in the above theorems, that the group G is finite or nilpotent, and that α : G −→ G is a fix-point-free automorphism of finite order n. Then 2 n 1 2 n 1 the transformation 1+α+α +···+α − of G, defined by x 7→ x·α(x)·α (x) ··· α − (x), vanishes identically. If, moreover, n is a prime, then the group G will have no n-torsion. In view of these impressive results, we propose the following family of problems. d Meta-Problem 1.1.5. We are given a polynomial r(t) := a0 + a1 · t + ··· + ad · t with integer coefficients, and we are told that some finite group G admits a fix-point-free automorphism α : G −→ G such that the map G −→ G, defined by x 7→ xa0 · α(xa1 ) · α2(xa2 ) ··· αd(xad ), vanishes identically. Prove that the group G has a “good structure” modulo “bad torsion.” The main objective of this text is to refine theorem 1.1.1 and to generalize theorems 1.1.2 and 1.1.3 in the spirit of Meta-Problem 1.1.5. We will do this in theorem 1.2.8, theorem 1.2.4, and theorem 1.2.5 respectively. This will then allow us to generalize other classic results of Hughes—Thompson, Kegel and E. Khukhro on the structure of finite groups with a p-splitting automorphism. We will do this in corollary 7.2.5 and corollary 7.2.6. In order to state our results in full generality, we will have to introduce some notation and terminology. But we can already formulate a corollary that captures the spirit of our main results. d Corollary 1.1.6 (Irreducible case). Let r(t) = a0 + a1 · t + ··· + ad · t ∈ Z[t] be a polynomial that is irreducible over Q. Then there exist constants a,b,c ∈ N with the following property. Consider a finite group G with a fix-point-free automorphism α : G −→ G and suppose that for all x ∈ G we have the equality a0 a1 2 a2 d ad x · α(x ) · α (x ) ··· α (x )=1G. Then G is solvable and of the form A · (B ⋊ (C × D)), where A is an a-group, B is a d b-group, C is a nilpotent c-group, and D is a nilpotent group of class at most d2 . In particular: • If G has no (a · b)-torsion, then G is nilpotent. d • If G has no (a · b · c)-torsion, then G is nilpotent of class at most d2 . Here, we recall terminology from [4]. Let n ∈ Z. A group H is said to be an n-group if the order of every element of H divides some natural power of n. We say that a group H n has n-torsion if some h ∈ H \{1H } satisfies h =1H . Otherwise, we say that H has no n-torsion. This naturally generalizes the notion of p-groups and p-torsion (for p a prime). 3 1.2 Main results 1.2.1 Invariants of good identities Definition 1.2.1 (Identities). We consider a group (G, ·), together with an endomorphism d γ : G −→ G, and a polynomial r(t) := a0 + a1 · t + ··· + ad · t ∈ Z[t]. We say that r(t) is a monotone identity of γ if and only if the map r(γ): G −→ G : x 7−→ xr(γ) := xa0 · a1 d ad γ(x ) ··· γ (x ) vanishes identically. In this case, we will simply write r(γ)=1G. More generally, we say that r(t) is an identity of γ if and only if there exists a decomposition r(t)= r1(t)+ ··· + rk(t) of r(t) into polynomials r1(t),...,rk(t) ∈ Z[t], such that the map r1(γ) rk (γ) r1(γ) ··· rk(γ): G −→ G : x 7−→ x ··· x vanishes identically. We will abbreviate this to r1(γ) ··· rk(γ)=1G. Definition 1.2.2 (Good polynomials). A polynomial r(t) ∈ Z[t] is said to be bad if and only if there exists some u ∈ N and some s(t) ∈ Z[t] \ Z such that s(tu+1) | r(0) · r(1) · r(t) in the ring Z[t]. Otherwise, we say that r(t) is good. For each polynomial r(t) ∈ Z[t], we will eventually introduce integer invariants Cong(r(t)), Discr (r(t)), and Prod(r(t)).