Narrow Results

Let $G$ be a finite group and $\sigma$=$\{{\sigma }_i|i\in I\}$ some partition of the set of all primes. A subgroup $H$ of $G$ is said to be $weakly$$\sigma$-$normal$ in $G$ if there exists a $\sigma$-subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\le H_{\ast }\le H$, where $H_{\ast }$ is either $\sigma$-permutably embedded or $\sigma$-semipermutable in $G$. In this paper, we investigate the influence of weakly $\sigma$-normal subgroups on the structure of finite groups.

Let $G$ be a finite group. An element $x$ in $G$ is termed as a vanishing element if there exists at least one irreducible character $\chi$ of $G$ such that $\chi (x)=0$. In this paper, we prove that if the set of vanishing elements of prime power orders in $G$ is the union of at most three conjugacy classes, then $G$ is solvable.

In this paper, we are interested in the relationships between the structure of a finite group $G$ and the zeros of strongly monolithic characters of $G$. We give some criteria for solvability and supersolvability of finite groups.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be an $\mathcal{\mathscr{H}}$-subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$. A subgroup $H$ of $G$ is called weakly $\mathcal{\mathscr{H}}$-subgroup in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is an $\mathcal{\mathscr{H}}$-subgroup in $G$. The aim of this paper is to analyze the structure of a finite group in which the fewer maximal subgroups of Sylow $p$-subgroup are weakly $\mathcal{\mathscr{H}}$-subgroups. Our results improve and generalize many known results.

Discretized non-Abelian gauge theories living on finite group spaces G are defined by means of a geometric action ∫ Tr F ∧ _*F. This technique is extended to obtain discrete versions of the Born–Infeld action. The discretizations are in 1–1 correspondence with differential calculi on finite groups.

A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a four-dimensional Abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations.

Yang–Mills and Born–Infeld theories are also considered on product spaces M^D×G, and we find the corresponding field theories on M^D after Kaluza–Klein reduction on the G discrete internal spaces. We examine in detail the case G=Z_N, and discuss the limit N→∞.

A self-contained review on the noncommutative differential geometry of finite groups is included.

We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S₃, the dihedral group D₄ and the quaternion group Q. Poincaré duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general.

A short review of the bicovariant (noncommutative) differential calculus on finite G is given for selfconsistency. Exterior derivative, exterior product, metric, Hodge dual, connections, torsion, curvature, and biinvariant integration can be defined algebraically. A projector decomposition of the braiding operator is found, and used in constructing the projector on the space of two-forms. By means of the braiding operator and the metric a knot invariant is defined for any finite group.

We give a list of all possible schemes for performing amino acid and codon assignments in algebraic models for the genetic code, which are consistent with a few simple symmetry principles, in accordance with the spirit of the algebraic approach to the evolution of the genetic code proposed by Hornos and Hornos. Our results are complete in the sense of covering all the algebraic models that arise within this approach, whether based on Lie groups/Lie algebras, on Lie superalgebras or on finite groups.

We give a full classification of the possible schemes for obtaining the distribution of multiplets observed in the standard genetic code by symmetry breaking in the context of finite groups, based on an extended notion of partial symmetry breaking that incorporates the intuitive idea of "freezing" first proposed by Francis Crick, which is given a precise mathematical meaning.

The paper discusses the Andrews–Curtis graph Δ_k(G,N) of a normal subgroup N in a group G. The vertices of the graph are k-tuples of elements in N which generate N as a normal subgroup; two vertices are connected if one of them can be obtained from another by certain elementary transformations. This object appears naturally in the theory of black box finite groups and in the Andrews–Curtis conjecture in algebraic topology [3].

We suggest an approach to the Andrews–Curtis conjecture based on the study of Andrews–Curtis graphs of finite groups, discuss properties of Andrews–Curtis graphs of some classes of finite groups and results of computer experiments with generation of random elements of finite groups by random walks on their Andrews–Curtis graphs.

We analyze the computational complexity of solving a single equation and checking identities over finite meta-abelian groups. Among others we answer a question of Goldmann and Russel from 1998: we prove that it is decidable in polynomial time whether or not an equation over the six-element group S₃ has a solution.

Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there exist finitely many finite groups $G$, up to isomorphism, having a normal subgroup $N$ of index $n$ which contains exactly $s$ non-central $G$-conjugacy classes. Upper bounds for the orders of $G$ and $N$ are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few $G$-classes. We also study the related problems when we consider only the set of $G$-classes of prime-power order elements contained in a normal subgroup.

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and $q$ are primes greater than $\frac{n}{3}$. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle $A_n$ if $n\le 17$.

We call a reduced word $w$multiplicity-bounding if and only if a finite group on which the word map of $w$ has a fiber of positive proportion $\rho$ can only contain each non-abelian finite simple group $S$ as a composition factor with multiplicity bounded in terms of $\rho$ and $S$. In this paper, based on recent work of Nikolov, we present methods to show that a given reduced word is multiplicity-bounding and apply them to give some nontrivial examples of multiplicity-bounding words, such as words of the form $x^e$, where $x$ is a single variable and $e$ an odd integer.

Let $G$ be a finite group and $I_G(\sigma )$ be the set of the elements $x$ of $G$ such that $\sigma (x)=x^{-1}$ where $\sigma \in Aut(G)$. In this paper, we give strong restrictions on the structure of the quotients $\frac{G}{O_2(G)}$ where $G$ is a finite group with an automorphism $\sigma$ such that $|I_G(\sigma )|\ge \frac{\left|G\right|}{3}$ and $O_2(G)$ is the largest normal $2$-subgroup of $G$.

A finite group $G$ is called a $\mathcal{𝒬}\mathcal{𝒮}$-group if every proper subgroup of $G$ is either quasi-normal or self-normal in $G$. In this paper, the authors classify the non-$\mathcal{𝒬}\mathcal{𝒮}$-groups whose proper subgroups are all $\mathcal{𝒬}\mathcal{𝒮}$-groups.

For a finite group $G$, let $\psi (G)$ denote the sum of element orders of $G$. If $n$ is a positive integer let $C_n$ be the cyclic group of order $n$. It is known that $\psi (C_n)$ is the maximum value of $\psi$ on the set of groups of order $n$, and $\psi (G)=\psi (C_n)$ if and only if $G$ is cyclic of order $n$. In this paper, we investigate the second largest value of $\psi$ on the set of groups of order $n$ and the structure of groups $G$ of order $n$ with this value of $\psi (G)$ when $n$ is odd.

We show that the number of homomorphisms from a knot group to a finite group G cannot be a Vassiliev invariant, unless it is constant on the set of (2, 2p+1) torus knots. In several cases, such as when G is a dihedral or symmetric group, this implies that the number of homomorphisms is not a Vassiliev invariant.

A subnormal subgroup X of G has the following property: there exists a Dirichlet polynomial Q(s) with integer coefficients such that, for each t ∈ ℕ, Q(t) is the conditional probability that t random elements generate G given that they generate G together with the elements of X In this paper we analyze how far can a subgroup X be with this property from being a subnormal subgroup.

Let 𝔉 be a class of finite groups. A subgroup H of G is said to be 𝔉-supplemented in G if there exists a subgroup T of G such that G = TH and (H ∩ T)H_G/H_G is contained in the 𝔉-hypercenter of G/H_G. In this paper, we investigate further the influence of 𝔉-supplemented subgroup on the structure of finite groups and generalize a series of known results.

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group C_n. In this paper, we investigate the minimum value of ψ on the set of groups of the same order.