Proper Subgroup Research Articles

On the maximal number of elements pairwise generating the symmetric group of even degree

Let G be the symmetric group of degree n. Let ω(G) be the maximal size of a subset S of G such that 〈x,y〉=G whenever x,y∈S and x≠y and let σ(G) be the minimal size of a family of proper subgroups of G whose union is G. We prove that both functions σ(G) and ω(G) are asymptotically equal to 12(nn/2) when n is even. This, together with a result of S. Blackburn, implies that σ(G)/ω(G) tends to 1 as n→∞. Moreover, we give a lower bound of n/5 on ω(G) which is independent of the classification of finite simple groups. We also calculate, for large enough n, the clique number of the graph defined as follows: the vertices are the elements of G and two vertices x,y are connected by an edge if 〈x,y〉≥An.

Finite groups for which all proper subgroups have consecutive character degrees

<abstract><p>Huppert and Qian et al. classified finite groups for which all irreducible character degrees are consecutive. The aim of this paper is to determine the structure of finite groups whose irreducible character degrees of their proper subgroups are all consecutive.</p></abstract>

Reduction of structure to parabolic subgroups

Let G be an affine group over a field of characteristic not two. A G -torsor is called isotropic if it admits reduction of structure to a proper parabolic subgroup of G . This definition generalizes isotropy of affine groups and involutions of central simple algebras. When does G admit anisotropic torsors? Building on work of J. Tits, we answer this question for simple groups. We also give an answer for connected and semisimple G under certain restrictions on its root system.

Additive bases of $C_3\oplus C_{3q}$

Let $G$ be a finite abelian group and $p$ be the smallest prime dividing $|G|$. Let $S$ be a sequence over $G$. We say that $S$ is regular if for every proper subgroup $H \subsetneq G$, $S$ contains at most $|H|-1$ terms from $H$. Let $\mathsf c_0(G)$ be

Nonsolvable Groups Whose Degrees of All Proper Subgroups Are the Direct Products of at Most Two Prime Numbers

Huppert and Manz have determined the nonsolvable groups whose character degrees are products of at most two prime numbers. In this paper, we change the condition from “degrees of a group are products of at most two prime divisors” to “degrees of all proper groups of a group are products of at most two prime divisors” and determine the structure of finite groups with such condition.

Explicit height bounds for K-rational points on transverse curves in powers of elliptic curves

Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper algebraic subgroups of $E^N$. The method gives a totally explicit version of the Manin-Dam'janenko Theorem in the elliptic case and it is a generalisation of previous results only proved when $E$ does not have Complex Multiplication.

First-order aspects of Coxeter groups

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the 2-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show to be essentially the Coxeter groups of affine type. Secondly, we characterize the Coxeter groups of finite rank which are domains, a central assumption in the theory of algebraic geometry over groups, which in many respects (e.g. λ-stability) reduces the model theory of a given Coxeter system to the model theory of its associated irreducible components. In the second part of the paper we move to specific definability questions in right-angled Coxeter groups (RACGs) and 2-spherical Coxeter groups. In this respect, firstly, we prove that RACGs of finite rank do not have proper elementary subgroups which are Coxeter groups, and prove further that reflection independent ones do not have proper elementary subgroups at all. Secondly, we prove that if the monoid Sim(W,S) of S-self-similarities of W is finitely generated, then W is a prime model of its theory. Thirdly, we prove that in reflection independent RACGs of finite rank the Coxeter elements are type-determined. We then move to 2-spherical Coxeter groups, proving that if (W,S) is irreducible, 2-spherical even and not affine, then W is a prime model of its theory, and that if WΓ and WΘ are as in the previous sentence, then WΓ is elementary equivalent to WΘ if and only if Γ≅Θ, thus solving the elementary equivalence problem for most of the 2-spherical Coxeter groups. In the last part of the paper we focus on model theoretic applications of the notion of reflection length from Coxeter group theory, proving in particular that affine Coxeter groups are not connected.

The Intersection Graph of Subgroups of the Dihedral Group of Order 2pq

For a finite group G, the intersection graph of G is the graph whose vertex set is the set of all proper non-trivial subgroups of G, where two distinct vertices are adjacent if their intersection is a non-trivial subgroup of G. In this article, we investigate the detour index, eccentric connectivity, and total eccentricity polynomials of the intersection graph of subgroups of the dihedral group for distinct primes . We also find the mean distance of the graph .

Numerical flatness and principal bundles on Fujiki manifolds

Let M be a compact connected Fujiki manifold, G a semisimple affine algebraic group over C with one simple factor and P a fixed proper parabolic subgroup of G. For a holomorphic principal G–bundle EG over M, let EP be the holomorphic principal P–bundle EG⟶EG/P given by the quotient map. We prove that the following three statements are equivalent: (1) ad(EG) is numerically flat, (2) the holomorphic line bundle ⋀topad(EP)⁎ is nef, and (3) for every reduced irreducible compact complex analytic space Z with a Kähler form ω, holomorphic map γ:Z⟶M, and holomorphic reduction of structure group EP⊂γ⁎EG to P, the inequality degree(ad(EP))≤0 holds.

Cantor dynamics of renormalizable groups

A group \Gamma is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding \varphi \colon \Gamma \to \Gamma whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for \Gamma .” In this work, we associate a dynamical system to a renormalization \varphi of \Gamma . The discriminant invariant {\mathcal D}_{\varphi} of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If {\mathcal D}_{\varphi} is a finite group for some renormalization, we show that \Gamma/C_{\varphi} is virtually nilpotent, where C_{\varphi} is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.

N-critical graph of finite groups

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.

Groups whose proper subgroups of infinite rank are Černikov-by-hypercentral or hypercentral-by-Černikov

Groups whose proper subgroups of infinite rank are Černikov-by-hypercentral or hypercentral-by-Černikov

Some results on the join graph of finite groups

‎Let $G$ be a finite group which is not cyclic of prime power order‎. ‎The join graph $Delta(G)$ of $G$ is a graph whose vertex set is the set of all proper subgroups of $G$‎, ‎which are not contained in the Frattini subgroup $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $G=langle H‎, ‎Krangle$‎. ‎Among other results‎, ‎we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $Delta(G)congDelta(H)$‎, ‎then $H$ is cyclic‎. ‎Also we prove that $Delta(G)congDelta(A_5)$ if and only if $Gcong A_5$‎.

Finite groups with three nonabelian subgroups

We characterize finite groups with exactly two nonabelian proper subgroups. When $G$ is nilpotent, we show that $G$ is either the direct product of a minimal nonabelian $p$-group and a cyclic $q$-group or a $2$-group. When $G$ is nonnilpotent supersolvable group, we obtain the presentation of $G$. Finally, when $G$ is nonsupersolvable, we show that $G$ is a semidirect product of a $p$-group and a cyclic group.

Groups whose proper subgroups are linear

The aim of this paper is to investigate groups whose proper subgroups are linear. Although there exist locally nilpotent non-linear groups in which every proper subgroup is linear, we prove that, for a large group class containing in particular all hypercentral groups, the linearity of the proper subgroups implies that of the whole group. Moreover, we study groups whose proper subgroups are linear of characteristic 0, showing in particular that these groups are themselves linear of characteristic 0 provided that they are either locally nilpotent or locally finite.

Nonsmoothable actions of [formula omitted] on spin four-manifolds

We construct some nonsmoothable actions of Z2×Z2 on spin four-manifolds by using an equivariant version of Furuta's 10/8-inequality. The examples satisfy following property: any proper subgroup of Z2×Z2 is smoothable for some smooth structure.

Expansions of Presburger arithmetic with the exchange property

AbstractLet G be a model of Presburger arithmetic. Let be an expansion of the language of Presburger . In this paper, we prove that the ‐theory of G is ‐minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the ‐theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H. Assuming that the induced theories on H and are definable complete and o‐minimal respectively, we prove that any definable set of G is ‐definable.

On the asymptotic enumeration of Cayley graphs

In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

On co-maximal subgroup graph of $Z_n$

The co-maximal subgroup graph $Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK = G$‎. ‎In this paper‎, ‎we study and characterize various properties like diameter‎, ‎domination number‎, ‎perfectness‎, ‎hamiltonicity‎, ‎etc‎. ‎of $Gamma(mathbb{Z}_n)$

Finite groups with formational subnormal strictly 2-maximal subgroups

Let be a subgroup-closed saturated Š-formation containing all nilpotent groups and let G be a group such that We prove that the requirement of -subnormality of all strictly 2-maximal subgroups coincides with the requirement of subnormality of all 2-maximal subgroups and is equivalent to G is a non-nilpotent group all of whose proper subgroups are abelian.