Lie Rings Research Articles

Frobenius groups as groups of automorphisms

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.

Fixed points of the complements of Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BA with kernel B and complement A. It is proved that if N is a BA-invariant normal subgroup of G such that (|N|, |B|) = 1 and C N (B) = 1 then C G/N (A) = C G (A)N/N. If N = G is a nilpotent group then we give as a corollary some description of the fixed points C L(G)(A) in the associated Lie ring L(G) in terms of C G (A). In particular, this applies to the case where GB is a Frobenius group as well (so that GBA is a 2-Frobenius group, with not necessarily coprime orders of G and A).

Nilpotency of finite groups with Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that CG(C) is abelian. It is proved that if B is abelian of rank at least two and \({[C_G(u), C_G(v),\dots,C_G(v)]=1}\) for any \({u,v\in B{\setminus}\{1\}}\), where CG(v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and CG(b) is nilpotent of class at most c for every \({b \in B{\setminus}\{1\}}\), then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.

On the support of the free Lie algebra: the Schutzenberger problems

Combinatorics M.-P. Schutzenberger asked to determine the support of the free Lie algebra L(Zm) (A) on a finite alphabet A over the ring Z(m) of integers mod m and all pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L(Zm) (A) in A* as the set of all words w such that m divides all the coefficients appearing in the monomials of l* (w), where l* is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l* (w) via the shuffle product, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = (v) over tilde (resp. if n is even and u = (v) over tilde), where (v) over tilde denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l(n) on lambda-tabloids, where lambda is a partition of n. Representing a word w in two letters by the subset I of [n] = \1, 2, ... , n\ that consists of all positions that one of the letters occurs in w, the computation of l* (w) leads us to the notion of the Pascal descent polynomial p(n)(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for p(n) (I) and show that if m inverted iota Sigma(i is an element of I)(1)(i-1) (n - 1 i - 1), then w lies in the support of L(Zm) (A).

Lie powers of relation modules for groups

Motivated by applications to abstract group theory, we study Lie powers of relation modules. The relation module associated to a free presentation G = F / N of a group G is the abelianization N a b = N / [ N , N ] of N, with G-action given by conjugation in F. The degree n Lie power is the homogeneous component of degree n in the free Lie ring on N a b (equivalently, it is the relevant quotient of the lower central series of N). We show that after reduction modulo a prime p this becomes a projective G-module, provided n > 1 and n is not divisible by p.

On the Ideals of a Lie Ring of Derivations

Let R be a commutative ring, and D a Lie subring and an R-submodule of Der(R) such that R is D-semiprime (or D-prime). We investigate the structure of the ideals of D as Lie rings. As a consequence, we give a necessary and sufficient condition for the ideals of D to be semiprime (or prime, respectively) Lie rings.

Finding central decompositions of p-groups

Polynomial-time algorithms are given to find a central decomposition of maximum size for a finite p-group of class 2 and for a nilpotent Lie ring of class 2. The algorithms use Las Vegas probabilistic routines to compute the structure of finite *-rings and also the Las Vegas C-MeatAxe. When p is small, the probabilistic methods can be replaced by deterministic polynomial-time algorithms. The methods introduce new group isomorphism invariants including new characteristic subgroups.

Non-associative Gröbner bases, finitely-presented Lie rings and the Engel condition, II

We give an algorithm for constructing a basis and a multiplication table of a finite-dimensional finitely-presented Lie ring. Secondly, we give relations that are equivalent to the n -Engel condition, and only have to be checked for the elements of a basis of a Lie ring. We apply this to construct the freest t -generator Lie rings that satisfy the n -Engel condition, for ( t , n ) = ( 2 , 3 ) , ( 3 , 3 ) , ( 4 , 3 ) , ( 2 , 4 ) .

Classical algebra: its nature, origins, and uses

Preface Part 1. Numbers and Equations. Lesson 1. What Algebra Is. 1. Numbers in disguise. 1.1. Classical and modern algebra. 2. Arithmetic and algebra. 3. The environment of algebra: Number systems. 4. Important concepts and principles in this lesson. 5. Problems and questions. 6. Further reading. Lesson 2. Equations and Their Solutions. 1. Polynomial equations, coefficients, and roots. 1.1. Geometric interpretations. 2. The classification of equations. 2.1. Diophantine equations. 3. Numerical and formulaic approaches to equations. 3.1. The numerical approach. 3.2. The formulaic approach. 4. Important concepts and principles in this lesson. 5. Problems and questions. 6. Further reading. Lesson 3. Where Algebra Comes From. 1. An Egyptian problem. 2. A Mesopotamian problem. 3. A Chinese problem. 4. An Arabic problem. 5. A Japanese problem. 6. Problems and questions. 7. Further reading. Lesson 4. Why Algebra Is Important. 1. Example: An ideal pendulum. 2. Problems and questions. 3. Further reading. Lesson 5. Numerical Solution of Equations. 1. A simple but crude method. 2. Ancient Chinese methods of calculating. 2.1. A linear problem in three unknowns. 3. Systems of linear equations. 4. Polynomial equations. 4.1. Noninteger solutions. 5. The cubic equation. 6. Problems and questions. 7. Further reading. Part 2. The Formulaic Approach to Equations. Lesson 6. Combinatoric Solutions I: Quadratic Equations. 1. Why not set up tables of solutions?. 2. The quadratic formula. 3. Problems and questions. 4. Further reading. Lesson 7. Combinatoric Solutions II: Cubic Equations. 1. Reduction from four parameters to one. 2. Graphical solutions of cubic equations. 3. Efforts to find a cubic formula. 3.1. Cube roots of complex numbers. 4. Alternative forms of the cubic formula. 5. The \irreducible case. 5.1. Imaginary numbers. 6. Problems and questions. 7. Further reading. Part 3. Resolvents. Lesson 8. From Combinatorics to Resolvents. 1. Solution of the irreducible case using complex numbers. 2. The quartic equation. 3. Vi ete's solution of the irreducible case of the cubic. 3.1. Comparison of the Viete and Cardano solutions. 4. The Tschirnhaus solution of the cubic equation. 5. Lagrange's reflections on the cubic equation. 5.1. The cubic formula in terms of the roots. 5.2. A test case: The quartic. 6. Problems and questions. 7. Further reading. Lesson 9. The Search for Resolvents. 1. Coefficients and roots. 2. A unified approach to equations of all degrees. 2.1. A resolvent for the cubic equation. 3. A resolvent for the general quartic equation. 4. The state of polynomial algebra in 1770. 4.1. Seeking a resolvent for the quintic. 5. Permutations enter algebra. 6. Permutations of the variables in a function. 6.1. Two-valued functions. 7. Problems and questions. 8. Further reading. Part 4. Abstract Algebra. Lesson 10. Existence and Constructibility of Roots. 1. Proof that the complex numbers are algebraically closed. 2. Solution by radicals: General considerations. 2.1. The quadratic formula. 2.2. The cubic formula. 2.3. Algebraic functions and algebraic formulas. 3. Abel's proof. 3.1. Taking the formula apart. 3.2. The last step in the proof. 3.3. The verdict on Abel's proof. 4. Problems and questions. 5. Further reading. Lesson 11. The Breakthrough: Galois Theory. 1. An example of a solving an equation by radicals. 2. Field automorphisms and permutations of roots. 2.1. Subgroups and cosets. 2.2. Normal subgroups and quotient groups. 2.3. Further analysis of the cubic equation. 2.4. Why the cubic formula must have the form it does. 2.5. Why the roots of unity are important. 2.6. The birth of Galois theory. 3. A sketch of Galois theory. 4. Solution by radicals. 4.1. Abel's theorem. 5. Some simple examples for practice. 6. The story of polynomial algebra: a recap. 7. Problems and questions. 8. Further reading. Epilogue: Modern Algebra. 1. Groups. 2. Rings. 2.1. Associative rings. 2.2. Lie rings. 2.3. Special classes of rings. 3. Division rings and fields. 4. Vector spaces and related structures. 4.1. Modules. 4.2. Algebras. 5. Conclusion. Appendix: Some Facts about Polynomials. Answers to the Problems and Questions. Subject Index. Name Index.

Graded Lie rings with many commuting components and an application to 2-Frobenius groups

The following variations of the theorems of Higman, Kreknin, and Kostrikin are proved: let L be a (/n)-graded Lie ring with trivial zero-component L0 = 0; if for some m each grading component commutes with all but at most m components, then L is soluble of derived length bounded above in terms of m; if, in addition, n is a prime, then L is nilpotent of class bounded above in terms of m. As an application to 2-Frobenius groups, it is proved that if a finite Frobenius group BC with complement C of order t acts on a finite group A so that AB is also a Frobenius group, (t, A) = 1, and CA(C) is abelian, then A is nilpotent of class bounded above in terms of t. © 2008 London Mathematical Society.

A limit approach to group homology

In this paper, we consider for any free presentation G = F / R of a group G the coinvariance H 0 ( G , R a b ⊗ n ) of the nth tensor power of the relation module R a b and show that the homology group H 2 n ( G , Z ) may be identified with the limit of the groups H 0 ( G , R a b ⊗ n ) , where the limit is taken over the category of these presentations of G. We also consider the free Lie ring generated by the relation module R a b , in order to relate the limit of the groups γ n R / [ γ n R , F ] to the n-torsion subgroup of H 2 n ( G , Z ) .

A partial extension of Lazard's correspondence for finite $p$-groups

M. Lazard established a correspondence between finite p -groups of nilpotence class less than p and finite nilpotent Lie rings of p -power order and nilpotence class less than p . This correspondence has had many applications, but cannot generally be extended to p -groups of class p or larger. However, in this paper, we obtain a partial extension of Lazard's result for a p -group that is a product of normal subgroups of class less than p .

Zeta functions of Lie rings of upper-triangular matrices

We prove a two-part theorem on local ideal zeta functions of Lie rings of upper-triangular matrices. First, we prove that these local zeta functions display a strong uniformity. Secondly, we prove that these zeta functions satisfy a local functional equation. Some explicit examples of these zeta functions are also presented. Finally, we consider certain quotients of these Lie rings, showing that the strong uniformity continues to hold, and that under certain circumstances the functional equation does too.

Pro-finite p-adic Lie algebras

Let p be a prime number. A finite nilpotent Lie ring of characteristic a power of p is called finite- p. A pro- p Lie ring is an inverse limit of finite- p Lie rings. Pro- p Lie rings play a role in Lie theory similar to that played by pro- p groups in group theory. Every pro- p Lie ring admits the structure of a Lie algebra over the p-adic integers; furthermore, every p-adic Lie algebra of finite rank as a p-adic module has an open pro- p subalgebra. We make a detailed study of pro- p Lie rings in terms of various properties, including their topology, Prüfer rank, subring growth, and p-adic module structure. In particular, we prove the equivalence of the following conditions for a finitely generated pro- p Lie ring L: L has finite Prüfer rank; L is isomorphic to a closed subring of gl ( V ) for some p-adic module V of finite rank; and, for sufficiently large n, the Lie F p -subalgebra W n = 〈 e 12 , t e 22 〉 ⊆ gl 2 ( F p [ t ] / 〈 t n 〉 ) is not an open section of L. By reducing to the pro- p Lie ring case, we also prove that all Engelian pro-finite Lie rings are locally nilpotent. This is a Lie theoretic analogue of Zelmanov's theorem which states that every periodic pro- p group is locally finite.

Computational Group Theory

The workshop Computational Group Theory was the fifth of this title held at Oberwolfach. It was attended by 54 participants with broad geographic representation from four continents. Among the participants were 10 young researchers, visiting Oberwolfach for the first time, some of them still graduate students. We are grateful for the special EU funds making their visits possible. The lecture program was divided into two sections. In the first we had a series of four invited longer lectures, 50 minutes each, given by distinguished, selected participants, surveying recent major developments in their fields. The first of these lectures was held by Eamonn O'Brien (Auckland), who gave a talk on “The latest developments in the matrix groups computation project”. The second was given by Bill Kantor (Eugene) on “A presentation of presentations”, introducing the remarkable result that all finite simple groups (except possibly ^2G_2(3^{2k+1}) ) can be presented by a bounded number of generators and relations. In the third talk in this series Michael Vaughan-Lee (Oxford) reported on the recently completed classification of all p -groups of orders p^6 and p^7 . Finally, Gregor Kemper (München) surveyed new developments in Computational Invariant Theory. The other section was made up of 37 short twenty-minute lectures on new work or work in progress. We aimed at giving preference to the younger participants to present their results. This program structure prompted positive feedback by many participants. The short talks reported on various computational aspects – data structures, algorithms, complexity, computer experiments – in a broad range of topics, including matrix groups, p -groups, finitely presented groups, permutation groups, representation theory of groups, invariant theory, group cohomology, Lie algebras and combinatorics. The analysis of algorithms requires a thorough theoretical background in the respective fields. The talks also revealed the close interrelationship between the different topics. Work for finite p -groups uses Lie rings and algebraic groups, matrix group algorithms rely on methods for finitely presented groups as well as on representation theory, work on permutation groups uses representation theoretic information. The computational solution of open problems such as the construction of Brauer character tables or the cohomology rings of sporadic groups usually requires the application of almost all the known techniques. Methods from other parts of computer algebra, e.g., Gröbner bases, come into play in computational cohomology theory as well as in the construction of matrix groups which are factor groups of specific finitely presented groups, e.g., Hurwitz groups. Despite the relatively large number of talks at this workshop, there was plenty of time for discussions. Needless to say that this time was well spent: numerous collaborations were continued and various others were started.

Prime Lie Rings of Derivations of Commutative Rings II

Let R be a 2-torsionfree commutative ring, and D a Lie subring and an R-submodule of Der(R) such that R is D-prime. Then any ideal of the Lie ring D is itself a prime Lie ring.

Graded Lie algebras with few nontrivial components

We prove that if a (ℤ/nℤ)-graded Lie algebra L = ↕i=0n−1Li has d nontrivial components Li and the null component L0 has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (ℤ/nℤ)-graded Lie rings L = ↕i=0n−1 with few nontrivial components Li if the null component L0 has finite order m. These results generalize Kreknin’s theorem on the solvability of the (ℤ/nℤ)-graded Lie rings L = ↕i=0n−1Li with trivial component L0 and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components Li. The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].

On a decomposition of the second cohomology of an abelian topological group

Let C be an abelian topological group ,A a trivial C-module.In this paper we show that the second cohomology of C with the 2-divisible coefficient is isomorphic to the direct sum of the second cohomology of a Lie ring c , whose underlying group is C, and the group of symmetric cocycles. Also there is an isomorphism between the categories of abelian topological groups of nilpotency class two,with a continuous section , and that of Lie rings of nilpotency class two. Mathematics Subject Classification: 22A05

Prime Lie rings of derivations of commutative rings in characteristic 2

Let R be a commutative associative ring with 1 and let Der ( R ) be the Lie ring of all derivations of R. Suppose that D is a Lie subring and an R-submodule of Der ( R ) . When R is D-prime, we give necessary and sufficient conditions for D to be Lie prime. Since results of this nature are already known for rings R of characteristic different from 2, what is really new here is the characteristic 2 case.

Prime Lie Rings of Derivations of Commutative Rings

Let R be a 2-torsion free commutative ring with identity, and δ a nonzero derivation of R such that R is δ-prime. Then Rδ is a prime Lie ring and any nonzero ideal of Rδ contains an ideal of the form Jδ where J is a nonzero δ-ideal of R.