Lie rings admitting an automorphism of order 4 with few fixed points. II

We consider a Lie ring (algebra) L that admits an automorphism φ of order 4 with a finite number m of fixed points (with a fixed-point subalgebra of finite dimension m). It is proved that L contains a subring S of m-bounded index in the additive group L (a subalgebra S of m-bounded codimension), which possesses a nilpotent ideal I of class bounded by some constant, such that the factor-ring S/I is nilpotent of class ≤2. As a consequence, it is proved that, under the same conditions, L has a subring G of m-bounded index in the additive group of L (a subalgebra G of m-bounded codimension), in which an ideal generated by the Lie subring [G, ?²]=«ng?g+g^{? 2} | g∈G»ng (the subalgebra [G, ?²]=«ng?g+g^{? 2} | g∈G»ng is an ideal in G which) is nilpotent of class bounded by some constant (and its factor-algebra G/[G, ?²] is nilpotent of class ≤2 with a derived algebra (square) of m-bounded dimension). In proofs, we use the results of [1] and develop further the version of the method of generalized centralizers employed therein.     

On the action of derivations on nilpotent ideals of associative algebras

Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F.     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Nilpotent injectors in alternating groups

AnN-Injector in an arbitrary finite group is defined as a maximal nilpotent subgroup ofG containing a subgroupA ofG of maximal order, satisfying class (A)≦2. In a previous paper theN-Injectors of Sym(n) were determined. In this paper we determine theN-Injectors of Alt(n), after having determined the set of all nilpotent subgroups,A, of Sym(n) of maximal order satisfying class(A)≦2. It is shown that the set ofN-Injectors of Alt(n) consists of a unique conjugacy class, and ifn≠9, it coincides with the set of the nilpotent subgroups of Alt(n) of maximal order.     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

Calculating the mislin genus for a certain family of nilpotent groups

It is known that the Mislin genus of a finitely generated nilpotent group N with finite commutator subgroup admits an abelian group structure. In this paper, we compute explicitly that structure under the following additional assumptions: The torsion subgroup TN is abelian, the epimorphism N→N/TN splits and all automorphisms of TN commute with cinjugation by elements of N. Among the groups satisfying these conditions are all nilpotent split extensions of a finite cyclic group by a finitely free abelian group. We further prove that the function M ? M × N^k­1 k ≥ 2, which is in general a surjective homomorphism from the genus of N onto the genus of N^k , is an isomorphism at least in an imporatnt special case. Applications to the study of non-cancellation phenomena in group theory are given.     

Classification of Finite Completions of a Class Locally D 8 Amalgams

Let 𝒜 = (A ₁, A ₁₂, A ₂) be a locally D ₈ amalgam with a finite completion G. Suppose that A ₁ ∈ Syl ₂(G). We show that under these conditions |A ₁| ≤2⁵, or N _{A 1} (A ₁₂) is Abelian. As applications of our results, we determine all the finite completions G, up to O(G), in the case where N _{A 1} (A ₁₂) is non-Abelian.     

Estimates for Derivatives of the Green Functions for Noncoercive Differential Operators on Homogeneous Manifolds of Negative Curvature

We consider the Green functions G for second-order noncoercive differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A=R ⁺. Using some probabilistic and analytic techniques we obtain estimates for derivatives of the Green functions G with respect to the N and A-variables.     

Rings with Commuting Nilpotents and Zero Divisors

We explore the consequences of certain commutativity hypotheses on a single nilpotent element or the set N of all nilpotent elements. We give several sufficient conditions for N to be an ideal. We present some nontrivial examples, including an example in which N is commutative (in fact, N ² = {0}) and N is not an ideal. Received: April 27, 2006. Revised: July 31, 2006.     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N?H = N^|Ω| ? H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.     

Engel Theorem for Novikov Algebras

We prove that, if A is left-nil Novikov algebra, then A ² is nilpotent.     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

Nilpotent injectors in symmetric groups

AnN-Injector in an arbitrary finite groupG is defined as a maximal nilpotent subgroup ofG, containing a subgroupA ofG of maximal order satisfying class(A)>=2. Among other results theN-Injectors of Sym(n) are determined and shown to consist of a unique conjugacy class of subgroups of Sym(n).     

Elementary abelian operator groups

Letp be a prime,n a positive integer. Suppose thatG is a finite solvablep'-group acted on by an elementary abelianp-groupA. We prove that ifC _G (ϕ) is of nilpotent length at mostn for every nontrivial element ϕ ofA and |A|≥p ⁿ⁺¹ thenG is of nilpotent length at mostn+1.     

The Jacobson radical of the Laurent series ring

For a large class of rings A, including all rings with right Krull dimension, it is proved that for every automorphism ϕ of the ring A, the Jacobson radical of the skew Laurent series ring A((x, ϕ)) is nilpotent and coincides with N((x, ϕ)), where N is the prime radical of the ring A. If A/N is a ring of bounded index, then the Jacobson radical of the Laurent series ring A((x)) coincides with N((x)). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 209–215, 2006.     

Infinitesimally PI radical algebras

The Golod-Shafarevich examples show that not every finitely generated nil algebraA is nilpotent. On the other hand, Kaplansky proved that every finitely generated nil PI-algebra is indeed nilpotent. We generalise Kaplansky’s result to include those algebras that are only infinitesimally PI. An associative algebraA is infinitesimally PI whenever the Lie subalgebra generated by the first homogeneous component of its graded algebra gr(A)=⊕ _t⩾1 A ⁱ /A ⁱ⁺¹ is a PI-algebra. We apply our results to a problem of Kaplansky’s concerning modular group algebras with radical augmentation ideal. The author is supported by NSERC of Canada.     

The Normalizer Property for Integral Group Rings of Some Finite Nilpotent-by-Nilpotent Groups

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ?(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.     

On existentially closed and generic nilpotent groups

Letn≧2 be an integer. We prove the following results that are known in casen=2: The upper and the lower central series of an existentially closed nilpotent group of classn coincide. A finitely generic nilpotent group of classn is periodic and the center of a finitely generic torsion-free nilpotent group of classn is isomorphic toQ ⁺, whereas infinitely generic nilpotent groups do not enjoy these properties. We determine the structure of the torsion subgroup of existentially closed nilpotent groups of class 2. Finally we give an algebraic proof that there exist 2^κ non-isomorphic existentially closed nilpotent groups of classn in cardinalityK ≧N ₀. Some results of this paper were contained in [6].