Characterization of 3-Rewritable Finite Nilpotent Groups

Let n be an integer greater than 1. A group G is said to be “ n-rewritable” whenever for every n elements x ₁,…,x _n of G, there exist distinct permutations τ, σ on the set {1,2,…, n} such that x _τ(1) ··· x _τ(n) = x _{σ (1)} ··· x _{σ (n)}. In this article, we complete the classification of 3-rewritable finite nilpotent groups and prove that a finite nilpotent group G is 3-rewritable if and only if G has an abelian subgroup of index 2 or the derived subgroup has order < 6.     

Some Properties of Bell Groups

For any integer n ≠ 0,1, a group G is said to be “n-Bell” if it satisfies the identity [x ⁿ ,y] = [x,y ⁿ ]. It is known that if G is an n-Bell group, then the factor group G/Z ₂(G) has finite exponent dividing 12n ⁵(n ? 1)⁵. In this article we show that this bound can be improved. Moreover, we prove that every n-Bell group is n-nilpotent; consequently, using results of Baer on finite n-nilpotent groups, we give the structure of locally finite n-Bell groups. Finally, we are concerned with locally graded n-Bell groups for special values of n.     

On Groups with Given Same-Order Types

On any group G, define g ～ h if g, h ∈ G have the same order. The set of sizes of the equivalence classes with respect to this relation is called the same-order type of G. In this article we prove that a group of the same-order type {1, n} is nilpotent and of the same-order type {1, m, n} is solvable.     

Krull Dimension and Deviation in Certain Parafree Groups

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) ? n, where μ(G) is the minimum possible number of generators of G.

Let G be fg; then Hom(G, SL ₂?) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer.     

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

On finite groups with non-subnormal subgroups

For a finite group G, let n(G) denote the number of conjugacy classes of non-subnormal subgroups of G. In this paper, we show that a finite group G satisfying n(G)≤2|π(G)| is solvable, and for a finite non-solvable group G, n(G) = 2|π(G)|+1 if and only if G?A₅.     

On Finite n-Acentralizer Groups

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.     

Nonsoluble length of finite groups with restrictions on Sylow subgroups

By 𝔛(n) we denote the variety of all groups satisfying the law [x,y]ⁿ≡1, that is, groups with commutators of order dividing n. Let p be a prime and G a finite group whose Sylow p-subgroups have normal series of length k all of whose quotients belong to 𝔛(n). We show that the non-p-soluble length λ_p(G) of G is bounded in terms of k and n only (Theorem 1.2). In the case where p is odd, a stronger result is obtained (Theorem 1.3).     

Automorphisms of Relatively Free Nilpotent Lie Algebras

Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N _Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL _n (?) and N _Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL _n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras.     

Maximal 2-Local Geometries for the Symmetric Groups

Abstract

The set 𝒩_max (G, T) consisting of all maximal 2-local subgroups of G = Sym(n) which contain T, a Sylow 2-subgroup of G, is investigated. In addition to determining the structure of the subgroups in 𝒩_max (G, T), the simplicial sets of maximal rank are classified.     

On the Grassmann modules for the unitary groups

Let V be 2n-dimensional vector space over a field 𝕂 equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let 𝕂₀ ? 𝕂 be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, …, 2n}, there exist natural representations of the groups G ? U(2n, 𝕂/𝕂₀) and H = G ∩ SL(V) ? SU(2n, 𝕂/𝕂₀) on the k-th exterior power of V. With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Some Results on p-Nilpotence and Supersolvability of Finite Groups

Let G be a finite group. A subgroup K of a group G is called an ?-subgroup of G if N _G (K) ∩ K ^x  ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P ₀ ? P ₁ ? ··· ? P _n  = P is called a maximal chain of P provided that |P _i : P _i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P ₀ ? P ₁ ? ··· ? P _i  ? ··· ? P _n  = P such that P _i  ? ?(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.     

Minimal sumsets in finite solvable groups

Given a group G and positive integers r,s≤|G|, we denote by μ_G(r,s) the least possible size of a product set AB={ab∣a∈A,b∈B}, where A,B run over all subsets of G of size r,s, respectively. While the function μ_G is completely known when G is abelian [S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, Journal of Algebra 287 (2005) 449-457], it is largely unknown for G non-abelian, in part because efficient tools for proving lower bounds on μ_G are still lacking in that case. Our main result here is a lower bound on μ_G for finite solvable groups, obtained by building it up from the abelian case with suitable combinatorial arguments. The result may be summarized as follows: if G is finite solvable of order m, then μ_G(r,s)≥μ_G^′(r,s), where G^′ is any abelian group of the same order m. Equivalently, with our knowledge of μ_G^′, our formula reads .One nice application is the full determination of the function μ_G for the dihedral group G=D_n and all n≥1. Up to now, only the case where n is a prime power was known. We prove that, for all n≥1, the group D_n has the same μ-function as an abelian group of order |D_n|=2n.     

On Finite Groups with Few Automorphism Orbits

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) ≤5, then G is isomorphic to one of the groups A₅, A₆, PSL(2, 7), or PSL(2, 8). We also consider the case when ω(G) = 6 and show that, if G is a nonsolvable finite group with ω(G) = 6, then either G ≈ PSL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≈ A₅.     

Divisibility and structure constraints on automorphism groups of almost perfect 1-factorizations of

Let G be a permutation group acting on a set with N elements such that every permutation with more than m fixed points is the identity. It is easy to verify that |G| divides N(N − 1) ··· (N − m). We show that if gcd(|G|, m!) = 1, then |G| divides (N − i)(N − j) for some i and j satisfying 0 ≤ i < j ≤ m. We use this to show that any almost perfect 1-factorization of K_2n has an automorphism group whose cardinality divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2 as long as n is odd. An almost perfect 1-factorization (or APOF) is a 1-factorization in which the union of any three distinct 1-factors is connected. This result contrasts with an example of an APOF on K₁₂ given by Cameron which has PSL(2, ℤ₁₁) as its automorphism group [with cardinality 12(11)(5)]. When n is even and the automorphism group is solvable, we show that either G acts vertex transitively and n is a power of two, or |G| divides 2n − 2^a for some integer a with 2^a dividing 2n, or else |G| divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2. We also give a number of structure results concerning these automorphism groups. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 355–380, 1998     

Average-case complexity and decision problems in group theory

We investigate the average-case complexity of decision problems for finitely generated groups, in particular, the word and membership problems. Using our recent results on “generic-case complexity”, we show that if a finitely generated group G has word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then the average-case complexity of the word problem of G is linear time, uniformly with respect to the collection of all length-invariant measures on G. This results applies to many of the groups usually studied in geometric group theory: for example, all braid groups B_n, all groups of hyperbolic knots, many Coxeter groups and all Artin groups of extra-large type.     

Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups

In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ?B? = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ? _n+1-cover if and only if n is even and G? (C ₂) ⁿ , where by a ? _m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n < 10 we find all pairs (m,p) (m > 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ?B? = PG(m,p).     

On Groups Admitting a Fixed-Point-Free Elementary 2-Group of Automorphisms

We give a short proof that if G is a finite group of derived length k and if G admits a fixed-point-free action of the elementary group of order 2 ⁿ , then G has a normal series of length n all of whose quotients are nilpotent of class bounded in terms of k and n only.     

A Remark on the Structure of N-Engel Groups

We observe that there exists a positive integer c(n) such that for every locally nilpotent n-Engel group G we have that G/Z _c(n)(G) is of n-bounded exponent. This strengthens a result of Burns and Medvedev.