Nontrivial Subgroup Research Articles

Representations of [formula omitted]-type combinatorial categories

In this paper we consider representations of certain combinatorial categories, including the poset D of positive integers and division, the Young lattice Y of partitions of finite sets, the opposite category of the orbit category Z of (Z,+) with respect to nontrivial subgroups, and the category CI of finite cyclic groups and injective homomorphisms. We describe explicit upper bounds for homological degrees of their representations, and deduce that finitely presented representations (resp., representations presented in finite degrees) over a field form abelian subcategories of the representation categories. We also give an explicit description for the category of sheaves over the ringed atomic site (Z,Jat,C_), and show that irreducible sheaves are parameterized by primitive roots of the unit.

Automorphism group of a local inclusion graph over the general linear group

The inclusion graph of a group G, written as I n ( G ) , is defined to be an undirected graph with all nontrivial subgroups of G as its vertices, and two distinct subgroups H, K of G are adjacent if either H ≤ K or K ≤ H . A local inclusion graph of G with respect to a given subgroup S, written as I n ( G , S ) , is an induced subgraph of I n ( G ) with vertex set consisting of all nontrivial subgroups of G properly containing S. Indeed, I n ( G ) can be viewed as the special case when S is the identity subgroup. Let GL n ( F ) be the general linear group consisting of all n × n invertible matrices over a field F and T n ( F ) the subgroup of all invertible lower triangular matrices in GL n ( F ) . In this article, an arbitrary automorphism of I n ( GL n ( F ) , T n ( F ) ) is determined and it is proved that the automorphism group of I n ( GL n ( F ) , T n ( F ) ) is isomorphic to the direct product of S n − 1 and S 2 , where S n − 1 is the symmetric group of degree n − 1 .

The influence of non-perfect code subgroups on the structure of groups

Let [Formula: see text] be a graph. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when [Formula: see text] is an independent set and every vertex of [Formula: see text] is adjacent to exactly one vertex in [Formula: see text]. Let [Formula: see text] = Cay(G, S) be a Cayley graph of a finite group [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when there exists a Cayley graph [Formula: see text] of [Formula: see text] such that [Formula: see text] is a perfect code of [Formula: see text]. Recently, groups whose set of all subgroup perfect codes forms a chain are classified. Also, groups with no proper nontrivial subgroup perfect code are characterized. In this paper, we generalize it and classify groups whose set of all non-perfect code subgroups forms a chain.

On the connectivity of the generating and rank graphs of finite groups

The generating graph encodes how generating pairs are spread among the elements of a group. For more than ten years it has been conjectured that this graph is connected for every finite group. In this paper, we give evidence supporting this conjecture: we prove that it holds for all but a finite number of almost simple groups and give a reduction to groups without non-trivial soluble normal subgroups. Let d(G) be the minimal cardinality of a generating set for G. When d(G)≥3, the generating graph is empty and the conjecture is trivially true. We consider it in the more general setting of the rank graph, which encodes how pairs of elements belonging to generating sets of minimal cardinality spread among the elements of a group. It carries information even when d(G)≥3 and corresponds to the generating graph when d(G)=2. We prove that it is connected whenever d(G)≥3, giving tools and ideas that may be used to address the original conjecture.

Symmetries of Fano varieties

Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛. We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group. We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety. For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds. Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family. Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.

Blocks whose defect groups are Suzuki 2-groups

We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.

Lattice isomorphisms of orthodox semigroups with no nontrivial finite subgroups

Lattice isomorphisms of orthodox semigroups with no nontrivial finite subgroups

Application of Clifford Algebra on Group Theory

The orthogonal operators defined as similarity transformations on Euclidean space E can also be considered as group actions on the Clifford Algebra. In this paper, we investigate the finite subgroup of Euclidian space E of Geometric Algebra over a finite dimension vector space E. The hierarchy of the finite subgroups of Clifford Algebra C(E) is depicted through the lattice structure and we discussed the group action of these subgroups on the vector space E. Further, we shall address the number of non-trivial finite subgroups, Normal subgroups, and subnormal series of the subgroup of Clifford Algebra C(E) constructed over the vector space E by performing group action over the subgroup of Clifford Algebra C(E).

Transitivity of normal subgroups of the mapping class groups on character varieties

We prove that the action of any non-trivial normal subgroup of the mapping class group of a closed surface of genus g\geqslant 2 is almost minimal on the character variety X(\pi_{1}\Sigma_{g},\mathrm{SU}_{2}) : the orbit of almost every point is dense.

Слойная конечность в некоторых группах

Infinite groups with finiteness conditions for an infinite system of subgroups are studied. Groups with a condition: the normalizer of any non-trivial finite subgroup is a layer-finite group or the normalizer of any non-trivial finite subgroup has a layer-finite periodic part are studied for beginning in the locally finite class of group, then in the class of periodic groups of Shunkov and finally in the class of Shunkov groups which are contain a strongly embedded subgroup with an almost layer-finite periodic part. The group $G$ is called the Shunkov group if for any prime $p$ and for every finite subgroups $H$ from $G$ any two conjugate elements of order $p$ from the factor-group $N_G(H)/H$ generate a finite subgroup. Results for almost layer-finite groups and groups with almost layer-finite periodic part are transferred to layer-finite groups and groups with layer-finite periodic part. New characterizations of layer-finite groups and groups with layer-finite periodic part are obtained.

The construction of fuzzy subgroups of groups with units modulo 20 and 21

The number of fuzzy subgroups (NFSGs) of the groups of units U20 and U21 are determining in this study. The (NFSGs) of U20 was found based on diagram and using the lemma which state that the (NFSGs) of G is equal to the number of chains (NC) on the subgroups lattice of G. Moreover, we explain that there are only two fuzzy subgroups for a prime order group. Lagrange’s theorem is more important in this work, from Lagrange’s theorem, there is no nontrivial subgroup of U21 since the order of the group is prime. We prove that, if P1 (μ) = U21, then we get μ1(x) = θ1, ∀x ∈ U21. Thus we have one fuzzy subgroup of P1 (μ) = U21. If P1 (μ) = {1}, then we obtain μ2(x) = θ1, ∀x ∈ {1} and μ2 (x) = θ2,∀x∈U21\{1}. Thus we have 2 fuzzy subgroups of U21. We came up with the numbers 21 and 2 respectively.

There are genus one curves violating Hasse principle over every number field

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.

Legendre pairs of lengthsℓ≡ 0 (mod 5)

AbstractBy assuming a type of balance for lengthℓ=87\ell =87and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for lengthℓ=85\ell =85, we find LPs of these lengths. We then study the power spectral density (PSD) values ofmmcompressions of LPs of length5m5m. We also formulate a conjecture for LPs of lengthsℓ≡0\ell \equiv 0(mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range≤200\le 200for which the existence question of LPs remains unsolved from 12 to 10.

Finite solvable groups with a nilpotent normal complement subgroup

Let [Formula: see text] be a finite solvable group and [Formula: see text] a non-normal core-free solvable subgroup of [Formula: see text]. We show that if the normalizer of any nontrivial normal subgroup of [Formula: see text] is equal to [Formula: see text], then [Formula: see text] has a nilpotent normal complement [Formula: see text] such that [Formula: see text] and [Formula: see text] is a Frobenius group.

Groups with three projective characters

Working over the field of complex numbers, the inequivalent irreducible projective representations of a finite group G with 2-cocycle α are considered. The greatest common divisor of the degrees of the α-characters of these representations is used to obtain information about the degrees of the irreducible αS -characters of a Sylow subgroup S of G. Suppose now that G has exactly three irreducible α-characters. Then the potential form that the three degrees can take is found if G has a nontrivial cyclic Sylow subgroup. However, the main result is to show that G is solvable when the degrees are of the form a, ra and ka with certain restrictions on r and k, most notably that r = 1 or k. This involves detailed analysis of the possible actions of G on the irreducible αN -characters of a normal subgroup N of G.

Profinite groups with abelian Sylow subgroups

We extend the definition of a finite -group to profinite groups and give a description of profinite -groups as a triple semidirect product of two prosoluble groups with a semisimple group, extending an old result of A. M. Broshi to the profinite case. We also prove that a profinite -group with finitely generated non-trivial Fitting subgroup is metabelian-by-(finite exponent). If, in addition, G is finitely generated then it is virtually metabelian polycyclic. Communicated by Mandi Schaeffer Fry

Finite Abelian groups with positive genus subgroup intersection graphs

The intersection graph of subgroups of a finite group [Formula: see text] is a graph whose vertices are all nontrivial subgroups of [Formula: see text] and in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The non-orientable genus of a graph [Formula: see text] is the smallest positive integer [Formula: see text] such that [Formula: see text] can be embedded on [Formula: see text] ([Formula: see text]), where [Formula: see text] and [Formula: see text] are the surface obtained from the sphere by attaching [Formula: see text] handles and the sphere with [Formula: see text] added crosscaps, respectively. In this paper, we classify all finite abelian groups whose non-oritentable genus of intersection graphs of subgroups are 1–3, respectively.

ON p-SOLVABILITY AND AVERAGE CHARACTER DEGREE IN A FINITE GROUP

AbstractAssume that G is a finite group, N is a nontrivial normal subgroup of G and p is an odd prime. Let $\mathrm{Irr}_p(G)=\{\chi \in \mathrm{Irr}(G) : \chi (1)=1~\mathrm{or}~ p \mid \chi (1)\}$ and $\mathrm{Irr}_p(G|N)=\{\chi \in \mathrm{Irr}_p(G) : N \not \leq \mathrm{ker}\,\chi \}$ . The average character degree of irreducible characters of $\mathrm{Irr}_p(G)$ and the average character degree of irreducible characters of $\mathrm{Irr}_p(G|N)$ are denoted by $\mathrm{acd}_p(G)$ and $\mathrm{acd}_p(G|N)$ , respectively. We show that if $\mathrm{Irr}_p(G|N) \neq \emptyset $ and $\mathrm{acd}_p(G|N) < \mathrm{acd}_p(\mathrm{PSL}_2(p))$ , then G is p-solvable and $O^{p'}(G)$ is solvable. We find examples that make this bound best possible. Moreover, we see that if $\mathrm{Irr}_p(G|N) = \emptyset $ , then N is p-solvable and $P \cap N$ and $PN/N$ are abelian for every $P \in \mathrm{Syl}_p(G)$ .

On an extension of Hall’s criterion

Let A and G be finite groups such that A acts coprimely on G by automorphisms, assume that G has a maximal A-invariant subgroup M that is a direct product of some isomorphic simple groups, we prove that if G has a non-trivial A-invariant normal subgroup N such that N ≤ M and every non-nilpotent maximal A-invariant subgroup K of G not containing N has index a prime or the square of a prime, then G is solvable.

Directional ergodicity, weak mixing and mixing for [formula omitted]- and [formula omitted]-actions

For a measure preserving Zd- or Rd-action T, on a Lebesgue probability space (X,μ), and a linear subspace L⊆Rd, we define notions of direction L ergodicity, weak mixing, and strong mixing. For Rd-actions, it is clear that these direction L properties should correspond to the same properties for the restriction of T to L. But since an arbitrary L⊆Rd does not necessarily correspond to a nontrivial subgroup of Zd, a different approach is needed for Zd-actions. In this case, we define direction L ergodicity, weak mixing, and mixing in terms of the restriction of the unit suspension T˜ to L, but also restricted to the subspace of L2(X˜,μ˜) perpendicular to the suspension direction. For Zd-actions, we show (as is more or less clear for Rd) that these directional properties are spectral properties. For weak mixing Zd- and Rd-actions, we show that directional ergodicity is equivalent to directional weak mixing. For ergodic Zd-actions T, we explore the relationship between direction L properties as defined via unit suspensions and embeddings of T in Rd-actions. Finally, the structure of possible sets of non-ergodic and non-weakly mixing directions is determined, and genericity questions are discussed.