Unique Minimal Normal Subgroup Research Articles

On minimal coverings and pairwise generation of some primitive groups of wreath product type

The covering number of a finite group [Formula: see text], denoted [Formula: see text], is the smallest positive integer [Formula: see text] such that [Formula: see text] is a union of [Formula: see text] proper subgroups. We calculate [Formula: see text] for a family of primitive groups [Formula: see text] with a unique minimal normal subgroup [Formula: see text], isomorphic to [Formula: see text] with [Formula: see text] divisible by [Formula: see text] and [Formula: see text] cyclic. This is a generalization of a result of Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.

Finite solvable groups with two imprimitive monolithic characters

Let [Formula: see text] be a finite group and let [Formula: see text] be an irreducible character of [Formula: see text]. If [Formula: see text] has a unique minimal normal subgroup, then [Formula: see text] is called monolithic. The character [Formula: see text] is said to be imprimitive if [Formula: see text] is induced from a character of a proper subgroup of [Formula: see text]. In this paper, we classify the finite solvable groups having exactly two imprimitive monolithic characters whose kernels intersect trivially.

Skew product groups for monolithic groups

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups.

On finite totally 2-closed groups

An abstract group $G$ is called totally 2-closed if $H = H^{(2),\Omega}$ for any set $\Omega$ with $G\cong H\leq\textrm{Sym}_\Omega$, where $H^{(2),\Omega}$ is the largest subgroup of symmetric group of $\Omega$ whose orbits on $\Omega\times\Omega$ are the same orbits of $H$. In this paper, we prove that the Fitting subgroup of a totally 2-closed group is a totally 2-closed group. We also conjecture that a finite group $G$ is totally 2-closed if and only if it is cyclic or a direct product of a cyclic group of odd order with a generalized quaternion group. We prove the conjecture in the soluble case, and reduce the general case to groups $G$ of shape $Z\cdot X$, with $Z = Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian

Bases of twisted wreath products

We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product G=Tk:P acting on its base group Ω=Tk, where T is some non-abelian simple group and P is some group acting transitively on k={1,…,k} with k⩾2. We prove that if G is primitive on Ω and P is quasiprimitive on k, then G has base size 2. We also prove that the proportion of pairs of points that are bases for G tends to 1 as |G|→∞ when G is primitive on Ω and P is primitive on k. Lastly, we determine the base size of any quasiprimitive group of twisted wreath type up to four possible values (and three in the primitive case). In particular, we demonstrate that there are many families of primitive groups of twisted wreath type with arbitrarily large base sizes.

Asal Dereceli Monolitik Karakterlere Sahip Sonlu Gruplar

Let G be a finite group. An irreducible character χ is called monolithic when the factor group G/ker⁡(χ) has unique minimal normal subgroup. In this paper, we prove that for the smallest prime q dividing the order of G if G has a faithful imprimitive monolithic character of degree q, then G becomes a nonabelian q-group or a Frobenius group with cyclic Frobenius complement whose order is q. Under certain conditions, we also classify finite groups in which their nonlinear irreducible characters are monolithic.

Four-Valent Oriented Graphs of Biquasiprimitive Type

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma,G)$ where $\Gamma$ is 4-valent, connected and $G$-oriented ($G$-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs $(\Gamma, G) \in\mathcal{OG}(4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $\Gamma$. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N \cong T^k$ for a simple group $T$ and $k\in \{1,2,4,8\}$. This provides a crucial step towards a general description of the long-studied family $\mathcal{OG}(4)$ in terms of a normal quotient reduction. We also give several methods for constructing pairs $(\Gamma, G)$ of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup $N$.

Point-primitive, line-transitive generalised quadrangles of holomorph type

Abstract Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set 𝒫 ${\mathcal{P}}$ of Γ, and transitively on the lines of Γ. We show that the primitive action of G on 𝒫 ${\mathcal{P}}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples Γ for which the action of G on 𝒫 ${\mathcal{P}}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup M and M is non-Abelian.

On the probability of generating a monolithic group

A group L is primitive monolithic if L has a unique minimal normal subgroup, N, and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L, given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N. Information about PL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.

Minimal generating sets of maximal size in finite monolithic groups

Denote by m(G) the largest size of a minimal generating set of a finite group G. We want to estimate the difference m(G) − m(G/N) in the case when N is the unique minimal normal subgroup of G.

Covering monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $sigma(G) < sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is monolithic, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.

Covering Certain Monolithic Groups with Proper Subgroups

Given a finite non-cyclic group G, call σ(G) the least number of proper subgroups of G needed to cover G. In this article, we give lower and upper bounds for σ(G) for G a group with a unique minimal normal subgroup N isomorphic to where n ≥ 5 and G/N is cyclic. We also show that σ(A 5≀C 2) = 57.

Probabilistic generation of finite groups with a unique minimal normal subgroups

Let L be a finite group with a unique minimal normal subgroup, say N. We study the conditional probability P L, N (d) that d randomly chosen elements of L generate, L given that they generate L modulo N. In particular, we prove that if d≥d(L), then P L, N (d)≥½. Several applications to general questions on the generation of finite and profinite groups are described.

Complements of the socle in monolithic groups

We show that if H is a finite group with a unique minimal normal subgroup N , which is not abelian, then the number of conjugacy classes of complements of N in H is strictly smaller than jN j.

Topology of order complexes of intervals in subgroup lattices

We conjecture that the order complex of an open interval in the subgroup lattice of a finite group has the homotopy type of a wedge of spheres and prove that if ( H, G) is a minimal counterexample to this conjecture then either G is almost simple or G= HN, where N is the unique minimal normal subgroup of G, N is non-Abelian and H∩ N=1.

On the probability of generating finite groups with a unique minimal normal subgroup

Assume that a finite group G has a unique minimal normal subgroup, say N . We prove that if the order of N is large enough then the following is true: If d randomly chosen elements generate G modulo N , then these elements almost certainly generate G itself.

On Prefrattini residuals

All groups considered in the sequel are finite. Let (ℭ and denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frattini chief factors [8]. The purpose of this article is to develop and expand a concept that arises naturally with the residuals for these formations, namely each G-chief factor is non-complemented (Frattini). With respect to a solid set X of maximal subgroups, these properties are generalized respectively to so-called X-parafrattini (X-profrattini) normal subgroups for which each type is closed relative to products. The relationships among the unique maximal normal subgroups that result from these products, the solid set of maximal subgroups X, X-prefrattini subgroups, and the residuals of formations are explored. This leads to a well-defined collected of formations, the partially nonsaturated formations, with properties analogous to those which are totally non-saturated. In the development, attention is given to a set of maximal subgroups which is the image of a solid function defined on all groups, a weaker condition than that of a solid set. A result of particular interest answers affirmatively the long-standing conjecture that a non-trivial nC-group G is solvable if and only if each G-chief factor is complemented by a maximal subgroup. This will force a critical re-examination of the classification problem for nC-groups. Since the article continues the investigations on finite groups initiated in [2], a familiarity with that article is assumed. All other notation and terminology is from [6]. If M is a maximal subgroup of a group G and G/C or e G(M) is a monolithic primitive group, i.e. a group with a unique minimal normal subgroup, then M is called a monolithic maximal subgroupof G.

Primitive permutation groups with a doubly transitive subconstituent

AbstractLet Gbe a primitive permutation group on a finite set Ω. We investigate the subconstitutents of G, that is the permutation groups induced by a point stabilizer on its orbits in Ω, in the cases where Ghas a diagonal action or a product action on Ω. In particular we show in these cases that no subconstituent is doubly transitive. Thus if G has a doubly transitive subconstituent we show that G has a unique minimal normal subgroup N and either N is a nonabelian simple group or N acts regularly on Ω: we investigate further the case where N is regular on Ω.

Finite Projective Planes that Admit a Strongly Irreducible Collineation Group

This paper studies how coding theory and group theory can be used to produce information about a finite projective plane π and a collineation group G of π.A new proof for Hering's bound on |G| is given in 2.5. Using the idea of coding theory developed in [9], a relation between two rows of the incidence matrix of π with respect to a tactical decomposition is obtained in 2.1. This result yields, among other things, some techniques in calculating |G|, and generalizes a result of Roth [16], [see 2.4 and 2.5].Hering [7] introduced the notion of strong irreducibility of G, that is, G does not leave invariant any point, line, triangle or proper subplane. He showed that if in addition G contains a non-trivial perspectivity, then there is a unique minimal normal subgroup of G. This subgroup is either non-abelian simple or isomorphic to the elementary abelian group Z3 × Z3 of order 9.

Finite Groups with a Unique Minimal Normal Subgroup

Finite Groups with a Unique Minimal Normal Subgroup