Nontrivial Subgroup Research Articles

Gluing of endopermutation modules

We introduce the notion of Frobenius-friendly categories and obtain the slash functors on these categories. Let G be a finite p-group. We give a local construction of the -stable category of a big enough Frobenius-friendly category of -modules: an endopermutation -module in gives rise, via slash functors, to a family of endopermutation -modules, where Q runs over non-trivial subgroups of G, together with certain isomorphisms. Conversely, the data of a suitable compatible family of endopermutation -modules comes from an endopermutation -module in unique up to isomorphism in the -stable category.

On invertible elements in reduced C⁎-algebras of acylindrically hyperbolic groups

Let G be an acylindrically hyperbolic group. We prove that if G has no non-trivial finite normal subgroups, then the set of invertible elements is dense in the reduced C⁎-algebra of G. The same result is obtained for finite direct products of acylindrically hyperbolic groups.

Local holomorphic mappings respecting homogeneous subspaces on rational homogeneous spaces

Let $G/P$ be a rational homogeneous space (not necessarily irreducible) and $x_0\in G/P$ be the point at which the isotropy group is $P$. The $G$-translates of the orbit $Qx_0$ of a parabolic subgroup $Q\subsetneq G$ such that $P\cap Q$ is parabolic are called $Q$-cycles. We established an extension theorem for local biholomorphisms on $G/P$ that map local pieces of $Q$-cycles into $Q$-cycles. We showed that such maps extend to global biholomorphisms of $G/P$ if $G/P$ is $Q$-cycle-connected, or equivalently, if there does not exist a non-trivial parabolic subgroup containing $P$ and $Q$. Then we applied this to the study of local biholomorphisms preserving the real group orbits on $G/P$ and showed that such a map extend to a global biholomorphism if the real group orbit admits a non-trivial holomorphic cover by the $Q$-cycles. The non-closed boundary orbits of a bounded symmetric domain embedded in its compact dual are examples of such real group orbits. Finally, using the results of Mok-Zhang on Schubert rigidity, we also established a Cartan-Fubini type extension theorem pertaining to $Q$-cycles, saying that if a local biholomorphism preserves the variety of tangent spaces of $Q$-cycles, then it extends to a global biholomorphism when the $Q$-cycles are positive dimensional and $G/P$ is of Picard number 1. This generalizes a well-known theorem of Hwang-Mok on minimal rational curves.

Finite groups in which the sylowizers of each non-trivial primary subgroups are conjugate

A subgroup S of a group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. In this paper, we investigate the structure of the group G in which the sylowizers of each non-trivial primary subgroups in G are conjugate and give the non-solvable chief factors of G.

Arithmeticity of discrete subgroups containing horospherical lattices

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.

Bielliptic quotient modular curves with N square-free

Let N≥1 be an square free integer and let WN be a non-trivial subgroup of the group of the Atkin-Lehner involutions of X0(N) such that the modular curve X0(N)/WN has genus at least two. We determine all pairs (N,WN) such that X0(N)/WN is a bielliptic curve and the pairs (N,WN) such that X0(N)/WN has an infinite number of quadratic points over Q.

Subgroup perfect codes in Cayley sum graphs

Let $$\Gamma $$ be a graph with vertex set V. If a subset C of V is independent in $$\Gamma $$ and every vertex in $$V\setminus C$$ is adjacent to exactly one vertex in C, then C is called a perfect code of $$\Gamma $$. Let G be a finite group and let S be a square-free normal subset of G. The Cayley sum graph of G with respect to S is a simple graph with vertex set G and two vertices x and y are adjacent if $$xy\in S$$. A subset C of G is called a perfect code of G if there exists a Cayley sum graph of G which admits C as a perfect code. In particular, if a subgroup of G is a perfect code of G, then the subgroup is called a subgroup perfect code of G. In this paper, we give a necessary and sufficient condition for a non-trivial subgroup of an abelian group with non-trivial Sylow 2-subgroup to be a subgroup perfect code of the group. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. As an application, we classify the abelian groups whose every non-trivial subgroup is a subgroup perfect code. Moreover, we determine all subgroup perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.

On the structure of groups with polynomial growth III

We show that a compactly generated locally compact group of polynomial growth having no non-trivial compact normal subgroups can be embedded as a co-compact subgroup into a semidirect product of a connected, simply connected, nilpotent Lie group and a compact group. There is also a uniqueness statement for this extension.

Intersection graphs of general linear groups

Let [Formula: see text] be a field and [Formula: see text] the general linear group of degree [Formula: see text] over [Formula: see text]. The intersection graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] is a finite field containing at least three elements, then the diameter [Formula: see text] is [Formula: see text] or [Formula: see text]. We also classify [Formula: see text] according to [Formula: see text]. In case [Formula: see text] is infinite, we prove that [Formula: see text] is one-ended of diameter [Formula: see text] and its unique end is thick.

Characterization of subgroup perfect codes in Cayley graphs

A subset C of the vertex set of a graph Γ is called a perfect code in Γ if every vertex of Γ is at distance no more than 1 to exactly one vertex of C. A subset C of a group G is called a perfect code of G if C is a perfect code in some Cayley graph of G. In this paper we give sufficient and necessary conditions for a subgroup H of a finite group G to be a perfect code of G. Based on this, we determine the finite groups that have no nontrivial subgroup as a perfect code, which answers a question by Ma, Walls, Wang and Zhou.

Solubility of Finite Generalized Frobenius with the Kernel of Odd Index

A finite group $$G$$ is said to be a generalized Frobenius group with kernel$$F$$, if $$F$$ is a proper nontrivial normal subgroup of $$G$$ and for every element $$Fx$$ of prime order of the quotient group $$G/F$$ the coset $$Fx$$ of the group $$G$$ over $$F$$ has only $$p$$-elements for some prime $$p$$ depending on $$x$$. This article considers generalized Frobenius groups with insoluble kernel. We prove that a quotient group of a generalized Frobenius group over its insoluble kernel is a 2-group.

Thickness of the subgroup intersection graph of a finite group

<abstract> <p>Let $ G $ be a finite group. The intersection graph of subgroups of $ G $ is a graph whose vertices are all non-trivial subgroups of $ G $ and in which two distinct vertices $ H $ and $ K $ are adjacent if and only if $ H\cap K\neq 1 $. In this paper, we classify all finite abelian groups whose thickness and outerthickness of subgroup intersection graphs are 1 and 2, respectively. We also investigate the thickness and outerthickness of subgroup intersection graphs for some finite non-abelian groups.</p> </abstract>

Planarity and fixing number of inclusion graph of a nilpotent group

The inclusion graph of a finite group [Formula: see text], written as [Formula: see text], is defined to be an undirected graph whose vertices are all nontrivial subgroups of [Formula: see text], and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. For a graph [Formula: see text] with vertex set [Formula: see text], a set of vertices [Formula: see text] is called a fixing set of [Formula: see text] if the only automorphism of [Formula: see text] that fixes every element in [Formula: see text] is the identity. The fixing number of [Formula: see text] is the smallest size of a fixing set of [Formula: see text]. In this paper, we determine the finite nilpotent groups whose inclusion graphs are planar. Moreover, using the technique of characteristic matrices, we characterize the fixing sets and give the exact value on the fixing number of the inclusion graphs for finite cyclic groups.

Analysis of normality of quotient subgroups on matrices integers modulo prime

A set of matrices integers modulo prime forms a finite group. This group has trivial and nontrivial subgroups. The trivial subgroup is normal and for a non-trivial subgroup the normal properties will be investigated. Furthermore, if a nontrivial subgroup is a normal subgroup, then a quotient subgroup can be constructed. This paper discusses the characteristics of normality of quotient subgroup on matrices integers modulo prime. By study of literatures, the order of group play an important role in this quotient subgroup. We derive some characteristics of normality of quotient subgroup. The result is that the order of subgroup determine on the normality of a quotient subgroup.

On Extensions of Semigroups and Their Applications to Toeplitz Algebras

The paper deals with the normal extensions of cancellative commutative semigroups and the Toeplitz algebras for those semigroups. By the Toeplitz algebra for a semigroup S one means the reduced semigroup C*-algebra C*(S). We study the normal extensions of cancellative commutative semigroups by the additive group ℤn of integers modulo n. Moreover, we assume that such an extension is generated by one element. We present a general method for constructing normal extensions of semigroups which contain no non-trivial subgroups. The Grothendieck group for a given semigroup and the group of all integers are involved in this construction. Examples of such extensions for the additive semigroup of non-negative integers are given. A criterion for a normal extension generated by an element to be isomorphic to a numerical semigroup is given in number-theoretic terms. The results concerning the Toeplitz algebras are the following. For a cancellative commutative semigroup S and its normal extension L generated by one element, there exists a natural embedding the semigroup C*-algebra C*(S) into C*(L). The semigroup C*-algebra C*(L) is topologically ℤn-graded. The results in the paper are announced without proofs.

Taketa’s inequality and finite groups with a normal Hall subgroup

Let G be a finite group. Isaacs and Seitz have conjectured that an upper bound for the derived length of G is the number of irreducible character degrees of G, whenever G is solvable. In this paper, we show that a minimal counterexample to the Isaacs-Seitz conjecture does not contain a non-trivial normal Hall subgroup. On the other hand, Isaacs and Knutson have conjectured that the inequality holds for all solvable normal groups N of G, where . We will prove that this conjecture holds whenever N is a normal Hall subgroup of G and N satisfies the Isaacs-Seitz conjecture.

Characterization of some families of simple groups by their intersection graphs

For a finite group G, the intersection graph of G denoted is an undirected graph such that its vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent, when In this paper we discuss about the existence of a leaf in the intersection graph of a finite group and specially in a nonabelian finite simple group. Also we determine some families of finite nonabelian simple groups which are characterizable by their intersection graphs.

Regular dessins d'enfants with dicyclic group of automorphisms

Let Gn be the dicyclic group of order 4n. We observe that, up to isomorphisms, (i) for n≥2 even there is exactly one regular dessin d'enfant with automorphism group Gn, and (ii) for n≥3 odd there are exactly two of them. Each of them is produced on well known hyperelliptic Riemann surfaces. We obtain that the minimal genus over which Gn acts purely-non-free is σp(Gn)=n (this coincides with the strong symmetric genus of Gn when n is even). For each of the triangular conformal actions, every non-trivial subgroup of Gn has genus zero quotient, in particular, that the isotypical decomposition, induced by the action of Gn, of its jacobian variety has only one component. We also study conformal/anticonformal actions of Gn, on closed Riemann surfaces, with the property that Gn admits anticonformal elements. It is known that Gn always acts on a genus one Riemann surface with such a property. We observe that the next genus σhyp(Gn)≥2 over which Gn acts in that way is n+1 for n≥2 even, and 2n−2 for n≥3 odd. We also provide examples of pseudo-real Riemann surfaces admitting Gn as the full group of conformal/anticonformal automorphisms.

Group Geometrical Axioms for Magic States of Quantum Computing

Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

Finite Groups Close to Frobenius Groups

We study finite nonsoluble generalized Frobenius groups; i.e., the groups G with a proper nontrivial normal subgroup F such that each coset Fx of prime order p, as an element of the quotient group G/F, consists only of p-elements. The particular example of such a group is a Frobenius group, given that F is the Frobenius kernel of G, and also the Camina group.