LOCAL NILPOTENCY IN VARIETIES OF GROUPS WITH OPERATORS

It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.

We prove that if a finite group G of rank r admits an automorphism I� of prime order having exactly m fixed points, then G has a I�-invariant subgroup of (r,m)-bounded index which is nilpotent of r-bounded class (Theorem 1). Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened. The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3). The general case reduces modulo available results to the case of finite p-groups. For reduction to Lie rings powerful p-groups are also used. For them a useful fact is proved which allows us to "glue together" nilpotency classes of factors of certain normal series (Theorem 2).

Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group admitting a fixed-point-free automorphism (i. e., leaving only the identity element fixed) of prime order must be nilpotent. However, non-nilpotent groups are known which admit fixedpoint-free automorphisms of composite order. In all these cases one notices that the groups in question are solvable. Although the sample is rather restricted, it is not too unnatural to ask whether the condition that a finite group admit such an automorphism is strong enough to force solvability of the group. This question is related to another problem, which seems. equally difficult, which asks whether a finite group containing a cyclic subgroup which is its own normalizer must be composite. In the present paper we shall prove that a group G possessiilg a fixedpoint-f ree automorphism of order 4 is solvable. Although many of the ideas used carry over to the case in which 4 has order pq, and especially 2q, our key lemmas use the fact that 4 has order 4 in a crucial way. The proof depends upon a theorem of Philip Ilall which asserts that a finite group G is solvable if for every factorization of o(G) into relatively prime numbers m and n, G contains a subgroup of order m. We show (Lemma 7) that a group G which has a fixed-point-free automorphism of order 4 satisfies the conditions of H all's theorem. Once we k-now that G is solvable it is not difficult to prove that its commutator subgroup is nilpotent (Theorem 2). This -fact was also observed by Thompson. Graham Higman has shown [3] that there is a bound to the class of a p-group P which possesses an automorphism p of prime order q without fixedpoints. This does not carry over to automorphisms of composite order, for at the end of the paper we give an example due to Thompson of a family of p-groups of arbitrary high class each of which admits a fixed-point-free automorphism of order 4.

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovacs’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1.

Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

A nilpotent quotient algorithm for graded Lie rings

In 1957, Higman showed that a Lie algebra admitting a fixed-point-free automorphism is nilpotent, and that an analogous result also holds for a finite soluble group. Two years later, Thompson proved that a finite group having a fixed-point-free automorphism of prime order is soluble, and consequently nilpotent. Generalizing that situation, a few years ago, Kharchenko set up a conjecture on the solubility of a Lie algebra L admitting an automorphism of prime order whose fixed points lie in the center of L. A similar conjecture applies also with finite groups. Here we affirm the latter for the case where the order p of an automorphism is equal to 2 and deny it for all p>3.

Let G be a finite group, and assume that G has an automorphism of order at least ρ ⁢ | G | {\rho|G|} , with ρ ∈ ( 0 , 1 ) {\rho\in(0,1)} . We prove that if ρ > 1 / 2 {\rho>1/2} , then G is abelian, and if ρ > 1 / 10 {\rho>1/10} , then G is solvable, whereas in general, the assumption implies [ G : Rad ( G ) ] ≤ ρ - 1.78 {[G:\operatorname{Rad}(G)]\leq\rho^{-1.78}} , where Rad ⁡ ( G ) {\operatorname{Rad}(G)} denotes the solvable radical of G. We also prove analogous results for a larger class of self-transformations of finite groups, so-called bijective affine maps. Furthermore, we provide two results of independent interest: an upper bound on element orders in the holomorph of a finite group, and that every bijective affine map of a finite semisimple group has a cycle of length equal to the order of the map, extending a theorem of Horoševskiĭ.

The original motive for studying Lie rings with Engel condition stemmed from Burnside's problem. The restricted Burnside hypothesis (that any group of p on q generators has finite lower central series [8]) could be established for prime p by showing that any Lie nil ring of a certain kind on q generators is actually nilpotent. It is not known whether this sufficient condition is also necessary; nevertheless, there are advantages in treating the problem within the context of Lie rings. Transition from the group problem to the Lie-ring problem is set up by the Magnus representation. This maps the q group generators into the q generators of the Lie ring, while the significance in the Lie ring of the group's having p appears in two known ways. One is that, since the Magnus representation takes products into sums, the Lie ring must have pf = 0 for every elementf. The other lies deeper, and requires for the Lie ring that if f and g are any elements of L then [f, gp-r ] = [... [f, g], g, . . . , g] = O (the bracket denotes Lie multiplication; there are p -1 g's on the right). Because Engel's theorem (for Lie algebras) begins with such an assumption, a condition like this is called an Engel condition of p (though some writers prefer exponent p -1 ). In actual computations it is not enough to know that each [f, gp-l] =0. The information needed is that all of what will here be called S,'s, the strongly homogeneous parts of the polynomials [f, gp-1], are also zero. Over a field, as in computations directly bearing on Burnside's problem, a standard Vandermonde determinant argument shows that indeed, if all [f, gP-l] = 0, then also all such S, = 0. When the scalars may not admit division this is no longer necessarily so. The important role of the Engel condition in such computations has led to creation of an independent area of study in groups and rings subjected to an Engel condition (see, e.g., [4; 6]). To study the condition by itself, apart from the other consequence (pf = 0) of assuming p in the Burnside group, one may begin with a

Lie rings with almost regular automorphisms

Lie centrally metabelian group rings

We show that if G F H GFH is a double Frobenius group with “upper” complement H H of order q q such that C G ( H ) C_G(H) is nilpotent of class c c , then G G is nilpotent of ( c , q ) (c,q) -bounded class. This solves a problem posed by Mazurov in the Kourovka Notebook. The proof is based on an analogous result on Lie rings: if a finite Frobenius group F H FH with kernel F F of prime order and complement H H of order q q acts on a Lie ring K K in such a way that C K ( F ) = 0 C_K(F)=0 and C K ( H ) C_K(H) is nilpotent of class c c , then K K is nilpotent of ( c , q ) (c,q) -bounded class.

A celebrated result of J. Thompson says that if a finite group \(G\) has a fixed-point-free automorphism of prime order, then \(G\) is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.