Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

Let G be a permutation group on a finite set . A sequence B=(₁,..., _b) of points in is called a base if its pointwise stabilizerin G is the identity. Bases are of fundamental importance incomputational algorithms for permutation groups. For both practicaland theoretical reasons, one is interested in the minimal basesize for (G, ), For a nonredundant base B, the elementary inequality2^|B||G|||^|B| holds; in particular, |B|log|G|/log||. In the casewhen G is primitive on , Pyber [8, p. 207] has conjectured thatthe minimal base size is less than Clog|G|/log|| for some (large)universal constant C. It appears that the hardest case of Pyber's conjecture is thatof primitive affine groups. Let H=GV be a primitive affine group;here the point stabilizer G acts faithfully and irreduciblyon the elementary abelian regular normal subgroup V of H, andwe may assume that =V. For positive integers m, let mV denotethe direct sum of m copies of V. If (v₁, ..., v_m)mV belongsto a regular G-orbit, then (0, v₁, ..., v_m) is a base for theprimitive affine group H. Conversely, a base (₁, ..., _b) forH which contains 0V= gives rise to a regular G-orbit on (b–1)V. Thus Pyber's conjecture for affine groups can be viewed asa regular orbit problem for G-modules, and it is therefore aspecial case of an important problem in group representationtheory. For a related result on regular orbits for quasisimplegroups, see [4, Theorem 6].     

On Regular Orbits of Elements of Classical Groups in Their Permutation Representations

Let H be a simple finite classical group not isomorphic to PSL(n, q) for any n, q. We prove that every cyclic subgroup of H has a regular orbit on every nontrivial permutation H-set.     

关于正则轨道问题的一个例子

本文给出了关于正则轨道问题的一个反例.     

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.     

可解群的大轨道

G可解群,G忠实的作用在有限群H上,且(|G|,|H|)=1,那么存在x,y∈H满足C_G(x)∩C_G(y)=1,即G有大轨道.     

Orbits of Inner Mapping Groups

 Several new tools are developed dealing with multiplication groups of loops. One of them is based upon the theorem that the inner mapping group of a loop never acts faithfully on any of its orbits, unless it is trivial. The applications include an analysis of loops whose inner mapping group is nonabelian of order pq. Some important earlier results are proved anew in a shorter way.     

On Orbits of Automorphism Groups

If G is a finite group and if A is a group of automorphisms of G whose fixed point subgroup is C _G (A) then every subgroup F of C _G (A) acts on the set of orbits of A in G. The peculiarities of this action are used here to derive several results on the number of orbits of A in an economical manner.Original Russian Text Copyright © 2005 Deaconescu M. and Walls G. L.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 533–537, May–June, 2005.     

Orbits of Inner Mapping Groups

 Several new tools are developed dealing with multiplication groups of loops. One of them is based upon the theorem that the inner mapping group of a loop never acts faithfully on any of its orbits, unless it is trivial. The applications include an analysis of loops whose inner mapping group is nonabelian of order pq. Some important earlier results are proved anew in a shorter way. (Received 5 March 2001; in revised form 11 June 2001)     

The Numbers of Permutation Groups Orbits and of the Wreath Product Orbits

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On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

On Orbits and Bi-invariant Subsets of Binary $$G$$ -Spaces

Mathematical Notes - Orbits and bi-invariant subsets of binary $$G$$ -spaces are studied. The problem of the distributivity of a binary action of a group $$G$$ on a space $$X$$ , which was posed in...     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

Solvable Permutation Groups and Orbits on Power Sets

We consider the class ? of finitely generated toral relatively hyperbolic groups. We show that groups from ? are commutative transitive and generalize a theorem proved by Benjamin Baumslag in [3 Baumslag, B. (1967). Residually free groups. Prceedings of the London Mathematical Society 17(3):402–418.[Crossref] , [Google Scholar]] to this class. We also discuss two definitions of (fully) residually-𝒞 groups, i.e., the classical Definition 1.1 and a modified Definition 1.4. Building upon results obtained by Ol'shanskii [18 Ol'shanskii, A. Yu. (1993). On residualing homomorphisms and G-subgroups of hyperbolic groups. International Journal of Algebra Computation 3:365–409.[Crossref] , [Google Scholar]] and Osin [22 Osin, D. V. (2010). Small cancellations over relatively hyperbolic groups and embedding theorems. Annals of mathematics 172:1–39.[Crossref], [Web of Science ®] , [Google Scholar]], we prove the equivalence of the two definitions for 𝒞 = ?. This is a generalization of the similar result obtained by Ol'shanskii for 𝒞 being the class of torsion-free hyperbolic groups. Let Γ ∈ ? be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [14 Kharlampovich, O., Myasnikov, A. (2012). Limits of relatively hyperbolic groups and Lyndon's completions. Journal of the European Math. Soc. 14:659–680.[Crossref], [Web of Science ®] , [Google Scholar]] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem that every finitely generated fully residually-Γ group embeds into a group from ?. On the other hand, we give an example of a finitely generated torsion-free fully residually-? group that does not embed into a group from ?; ? is the class of hyperbolic groups.     

On Subsets with Small Product in Torsion-Free Groups

G be a nonabelian torsion-free group. Let C be a finite generating subset of G such that . We prove that, for all subsets B of G with , we have . In particular, a finite subset X with cardinality satisfies the inequality if and only if there are elements , such that the following two conditions hold: (i) . (ii) where . Received: October 13, 1997/Revised: Revised August 18, 1998     

Primitive Rank 3 Groups on Symmetric Designs

We determine the symmetric designs which admit a group such that G has a nonabelian socle and is a primitiverank 3 group on points (and blocks).     

On Negative Orbits of Finite Coxeter Groups

For a Coxeter group W, X a subset of W and a positive root, we define the negative orbit of under X to be {w · | w X} ^–, where ^– is the set of negative roots. Here we investigate the sizes of such sets as varies in the case when W is a finite Coxeter group and X is a conjugacy class of W.     

Partitions of Groups into Large Subsets

Mathematical Notes -     

Regular Subsets of Semigroups Related to their Idempotents

For a subsemigroup T of a semigroup S, we let Reg(T) and reg(T) denote respectively the set of all regular elements of T and the set of all elements of T which are regular in S. We characterize semigroups with Reg(T)=reg(T), where T runs over one of the following families of subsemigroups: {Se | e ∈ E(S)}, {eS | e ∈ E(S)}, {eSf | e,f ∈ E(S)}.