Graded Lie Rings Research Articles

New Lie products for groups and their automorphisms

Abstract We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these rings to the automorphisms of a group. This process uncovers new structure in 4/5 of the approximately 11.8 million groups of size at most 1000 and beyond that point pertains to at least a positive logarithmic proportion of all finite groups.

Filters compatible with isomorphism testing

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring can be trivial or arbitrarily large. We determine properties of filters such that every isomorphism between groups is induced by an isomorphism between graded Lie rings.

On the stable Andreadakis problem

Let Fn be the free group on n generators. Consider the group IAn of automorphisms of Fn acting trivially on its abelianization. There are two canonical filtrations on IAn: the first one is its lower central series Γ⁎; the second one is the Andreadakis filtration A⁎, defined from the action on Fn. In this paper, we establish that the canonical morphism between the associated graded Lie rings L(Γ⁎) and L(A⁎) is stably surjective. We then investigate a p-restricted version of the Andreadakis problem. A calculation of the Lie algebra of the classical congruence group is also included.

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free d-generator Lie rings $$\mathfrak {f}_{c,d}$$ of nilpotency class c for all $$c\leqslant 2$$ and for $$(c,d)\in \{(3,3),(3,2),(4,2)\}$$ . We apply our computations to obtain information about $$\mathfrak {p}$$ -adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.

Efficient characteristic refinements for finite groups

Filters were introduced by J.B. Wilson in 2013 to generalize work of Lazard with associated graded Lie rings. It holds promise in improving isomorphism tests, but the formulas introduced then were impractical for computation. Here, we provide an efficient algorithm for these formulas, and we demonstrate their usefulness on several examples of p-groups.

Longer nilpotent series for classical unipotent subgroups

Abstract In studying nilpotent groups, the lower central series and other variations can be used to construct an associated ℤ+-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing ℕ d -graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field ℤ/pℤ of rank d. We prove that, for all the classical types, this characteristic filter is a series of length Θ(d 2) with nearly all factors having p-bounded order.

On groups and Lie algebras admitting a Frobenius group of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with kernel F and complement H such that CG(H) is nilpotent of class c. We prove that if F is abelian of rank at least two and [CG(a),CG(b),…,CG(b)]=1 for any a,b∈F∖{1}, where CG(b) is repeated k times, then G is nilpotent of class bounded in terms of c, k and |H| only. It is also proved that if F is abelian of rank at least three and CG(a) is nilpotent of class at most d for every a∈F∖{1}, then G is nilpotent of class bounded in terms of c, d and |H|. The proofs are based on results on graded Lie rings.

ON FOX QUOTIENTS OF ARBITRARY GROUP ALGEBRAS

Certain subquotients of group algebras are determined as a basis for subsequent computations of relative Fox and dimension subgroups. More precisely, for a group G and N-series [Formula: see text] of G let [Formula: see text], n ≥ 0, denote the filtration of the group algebra R(G) induced by [Formula: see text], and IR(G) its augmentation ideal. For subgroups H of G, left ideals J of R(H) and right H-submodules M of [Formula: see text] the quotients IR(G)J/MJ are studied by homological methods, notably for M = IR(G)IR(H), IR(H)IR(G) + I([H, G])R(G) and [Formula: see text] for a normal subgroup N in G; in the latter case the module IR(G)J/MJ is completely determined for n = 2. The groups [Formula: see text] are studied and explicitly computed for n ≤ 3 in terms of enveloping rings of certain graded Lie rings and of torsion products of abelian groups.

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.

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.

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 graphs and Lie rings

From a finite oriented graph Γ, finite-dimensional graded nilpotent Lie rings \(\mathfrak{l}\left( \Gamma \right)\)(Γ) and \(\mathfrak{g}\left( \Gamma \right)\)(Γ) are naturally constructed; these rings are related to subtrees and connected subgraphs of Γ, respectively. Diverse versions of these constructions are also suggested. Moreover, an embedding of Lie rings of the form \(\mathfrak{l}\left( \Gamma \right)\)(Γ) in the adjoint Lie rings of finite-dimensional associative rings (also determined by the graph Γ) is indicated.

Formula omitted]-graded Lie rings

Thompson proved that every finite group with a fixed point free automorphism of prime order is nilpotent and G. Higman proved that the nilpotency class is bounded in terms of the prime alone. Kreknin and Kostrikin produced the first explicit bound by reducing to the problem of bounding the nilpotency class of a Z p -graded Lie ring L with L 0 = 0 . Meixner later improved this bound. A step in the proof of Kreknin and Kostrikin is to bound the derived length of Z n -graded Lie rings L with L 0 = 0 . In this paper we improve these bounds.

On [formula omitted]-graded Lie rings

Let L be a Z 6 -graded Lie ring with L 0=0. We prove that L is soluble and the derived length of L is at most four. Moreover, γ 3( L) is nilpotent and the nilpotency class of γ 3( L) is at most six.

Some examples of homotopy Π-algebras

The homotopy Π-algebra of a pointed topological space, X, consists of the homotopy groups of X together with the additional structure of the primary homotopy operations. We extend two well-known results for homotopy groups to homotopy Π-algebras and look at some examples illustrating the depth of structure on homotopy groups; from graded group to graded Lie ring, to Π-algebra and beyond. We also describe an abstract Π-algebra and give three abstract Π-algebra structures on the homotopy groups of the loop space of X which can be realized as the homotopy Π-algebras of three different spaces.

Links, pictures and the homology of nilpotent groups

We give a geometric slice-like characterization for the vanishing of Milnor's link invariants by proving the k-slice conjecture. This conjecture states that a link L has vanishing Milnor μ ̄ -invariants of length ⩽2 k if and only if L bounds disjoint surfaces in a four disk in such a way that the fundamental group of the complement admits free nilpotent quotients of class k. In the course of our proof, we compute the dimension ⩽3 homology groups of finitely generated free nilpotent Lie rings and groups. We develop a new algorithm for constructing a weighted chain resolution for a nilpotent group with torsion free lower central series quotients, and with the property that its associated graded complex is the Koszul complex of the associated graded Lie ring. This give a new derivation of the May spectral sequence relating the group homology of the nilpotent group to the Lie ring homology of its associated graded Lie ring. Finally, we define μ ̄ -invariants of “pictures” and use these to describe a generating set of cocycles in the cohomology of the free nilpotent groups. Some sample computations follow.

On the derived length of finite, graded Lie rings with prime-power order, and groups with prime-power order

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Some Lie rings associated with Burnside groups

We describe some calculations in graded Lie rings which provide a fairly sharp upper bound for the nilpotency class and for the order of the restricted Burnside group on two generators with exponent 7.

A nilpotent quotient algorithm for graded Lie rings

A nilpotent quotient algorithm for graded Lie rings of prime characteristic is described. Thealgorithm has been implemented and applications have been made to the investigation of the associated Lie rings of Burnside groups. New results about Lie rings and Burnside groups are presented. These include detailed information on groups of exponent 5 and 7 and their associated Lie rings.

On lie gradings. I

The concept of a graded Lie ring L is studied systematically. A grading of L is defined as a decomposition of the additive group of L into a direct sum with simple multiplicative behavior of the components. To each grading Γ of L corresponds an abelian semigroup ▪ generated by the components. If L is a simple Lie algebra over the field F and the Γ-components are F -linear subspaces of L , then ▪ is a group. Its nonzero F -characters determine a diagonable subgroup Diag F Γ of the automorphism group of L over F . If, moreover, L is finite dimensional over F = C , then Γ is fine (cannot be refined) precisely if Diag F Γ is maximal diagonable in Aut C L .