Orbit-equivalent infinite permutation groups

Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.     

Abelian quotients and orbit sizes of solvable linear groups

Let G be a finite group, and let V be a completely reducible faithful Gmodule. It has been known for a long time that if G is abelian, then G has a regular orbit on V. In this paper we generalize this result as follows. Assuming G to be solvable, we show that G has an orbit of size at least |G/G′| on V. This also strengthens a result of Aschbacher and Guralnick in that situation. Additionally, we prove a similar generalization of the well-known result that if G is nilpotent, then G has an orbit of size at least \(\sqrt {\left| G \right|} \) on V.     

Extremely primitive groups and linear spaces

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let S be a nontrivial finite regular linear space and G ≤ Aut(S). Suppose that G is extremely primitive on points and let rank(G) be the rank of G on points. We prove that rank(G) ≥ 4 with few exceptions. Moreover, we show that Soc(G) is neither a sporadic group nor an alternating group, and G = PSL(2, q) with q + 1 a Fermat prime if Soc(G) is a finite classical simple group.     

Conjugacy class sizes and solvability of finite groups

Let G be a finite group and G?? be the set of primary, biprimary and triprimary elements of G. We prove that if the conjugacy class sizes of G?? are {1,m,n,mn} with positive coprime integers m and n, then G is solvable. This extends a recent result of Kong (Manatsh. Math. 168(2) (2012) 267–271).     

Bounding the Derived Length for a Given Set of Character Degrees

Let G be a finite solvable group with {1, a, b, c, ab, ac} as the character degree set, where a ,b, and c are pairwise coprime integers greater than 1. We show that the derived length of G is at most 4. This verifies that the Taketa inequality, dl(G) ≤ |cd(G)|, is valid for solvable groups with {1, a, b, c, ab, ac} as the character degree set. Also, as a corollary, we conclude that if a, b, c, and d are pairwise coprime integers greater than 1 and G is a solvable group such that cd(G) = {1, a, b, c, d, ac, ad, bc, bd}, then dl(G) ≤ 5. Finally, we construct a family of solvable groups whose derived lengths are 4 and character degree sets are in the form {1, p, b, pb, q ^p , pq ^p }, where p is a prime, q is a prime power of an odd prime, and b > 1 is integer such that p, q, and b are pairwise coprime. Hence, the bound 4 is the best bound for the derived length of solvable groups whose character degree set is in the form {1, a, b, c, ab, ac} for some pairwise coprime integers a, b, and c.     

A sufficient condition for nilpotency of the commutator subgroup

Let G be a finite group with the property that if a and b are commutators of coprime orders, then |ab| = |a||b|. We show that G′ is nilpotent.     

On a co-induction question of Kechris

This note answers a question of Kechris: if H < G is a normal subgroup of a countable group G, H has property MD and G/H is amenable and residually finite, then G also has property MD. Under the same hypothesis we prove that for any action a of G, if b is a free action of G/H, and b _G is the induced action of G, then CInd _H ^G (a|H) × b _G weakly contains a. Moreover, if H < G is any subgroup of a countable group G, and the action of G on G/H is amenable, then CInd _H ^G (a|H) weakly contains a whenever a is a Gaussian action.     

Irreducible disconnected systems in groups

LetG be an arbitrary group with a subgroupA. The subdegrees of (A, G) are the indices [A:A ∪A ⁹] (wheregεG). Equivalent definitions of that concept are given in [IP] and [K]. IfA is not normal inG and all the subdegrees of (A, G) are finite, we attach to (A, G) the common divisor graph Γ: its vertices are the non-unit subdegrees of (A, G), and two different subdegrees are joined by an edge iff they arenot coprime. It is proved in [IP] that Γ has at most two connected components. Assume that Γ is disconnected. LetD denote the subdegree set of (A, G) and letD ₁ be the set of all the subdegrees in the component of Γ containing min(D−{1}). We proved [K, Theorem A] that ifA is stable inG (a property which holds whenA or [G:A] is finite), then the setH={g ε G| [A:A ∪A ^g ] εD ₁ ∪ {1}} is a subgroup ofG. In this case we say thatA<H<G is a disconnected system (briefly: a system). In the current paper we deal with some fundamental types of systems. A systemA<H<G is irreducible if there does not exist 1<N△G such thatAN<H andAN/N<H/N<G/N is a system. Theorem A gives restrictions on the finite nilpotent normal subgroups ofG, whenG possesses an irreducible system. In particular, ifG is finite then Fit(G) is aq-group for a certain primeq. We deal also with general systems. Corollary (4.2) gives information about the structure of a finite groupG which possesses a system. Theorem B says that for any systemA<H<G,N _G (N _G (A))=N _G (A). Theorem C and Corollary C’ generalize a result of Praeger [P, Theorem 2]. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at Tel Aviv University under the supervision of Prof. Marcel Herzog.     

A polynomial-time algorithm for the paired-domination problem on permutation graphs

A set S of vertices in a graph H=(V,E) with no isolated vertices is a paired-dominating set of H if every vertex of H is adjacent to at least one vertex in S and if the subgraph induced by S contains a perfect matching. Let G be a permutation graph and π be its corresponding permutation. In this paper we present an O(mn) time algorithm for finding a minimum cardinality paired-dominating set for a permutation graph G with n vertices and m edges.     

On finite groups with exactly seven element centralizers

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Automorphisms of finite groups of bounded rank

LetG be a finite group admitting an automorphismα withm fixed points. Suppose every subgroup ofG isr-generated. It is shown that (1)G has a characteristic soluble subgroupH whose index is bounded in terms ofm andr, and (2) if the orders ofα andG are coprime, then the derived length ofH is also bounded in terms ofm andr. To Professor John Thompson, in honor of his outstanding achievements     

Base sizes for S-actions of finite classical groups

Let G be a permutation group on a set Ω. A subset B of Ω is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by b(G). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher’s collection S of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of Ω form a base for G. Indeed, with some specific exceptions, we show that almost all pairs of points in Ω are bases.     

A mass formula for self-dual permutation codes

For a module V over a finite semisimple algebra A we give the total number of self-dual codes in V. This enables us to obtain a mass formula for self-dual codes in permutation representations of finite groups over finite fields of coprime characteristic.     

On the number of relations in free products of abelian groups

We consider the finitely generated groups constructed from cyclic groups by free and direct products and study the question of the smallest number of relations for a given system of generators. This question is related to the relation gap problem. We prove that if m and n are not coprime then the group H _m,n = (? _m × ?) * (? _n × ?) cannot be defined using three relations in the standard system of generators. We obtain a similar result for the groups G _m,n = (? _m × ? _m ) * (? _n × ? _n ). On the other hand, we establish that for coprime m and n the image of H _m,n in every nilpotent group is defined using three relations.     

Base sizes for sporadic simple groups

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) ⩽ 7, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. This settles a conjecture of Cameron.     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

On the Finite Group with the Index of Maximal Subgroups of the Monster

Abstract

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or A _s if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s.     

Minimal permutation representations of finite simple orthogonal groups

In the paper, nontrivial permutation representations of minimal degree are studied for finite simple orthogonal groups. For them, we find degrees, ranks, subdegrees, point stabilizers and their pairwise intersections.Translated fromAlgebra i Logika, Vol. 33, No. 6, pp. 603–627, November–December, 1994.     

Large orbits of subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be a nilpotent subgroup of G; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and 3 ? |H|; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and assume the Sylow 2-subgroups of the semidirect product HV are abelian; then H has at least two regular orbits on V ⊕ V.