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In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices ${\rm L}(G)$ and sublattices of ${\rm L}(G)$. It is proved that the collections of all pronormal subgroups of ${\rm A}_n$ and S$_n$ do not form sublattices of respective ${\rm L}({\rm A}_n)$ and ${\rm L}({\rm S}_n)$, whereas the collection of all pronormal subgroups ${\rm LPrN}({\rm Dic}_n)$ of a dicyclic group is a sublattice of ${\rm L}({\rm Dic}_n)$. Furthermore, it is shown that ${\rm L}({\rm Dic}_n)$ and ${\rm LPrN}({\rm Dic}_n$) are lower semimodular lattices.

In the framework of weak congruence lattices, many classes of groups have been characterized up to now, in completely lattice-theoretic terms. In this note, the center of the group is captured lattice-theoretically and nilpotent groups are characterized by lattice properties. [ABSTRACT FROM AUTHOR]

In this paper, the concept of μF-subgroup is introduced. It is proved that in finite cyclic groups 풞n, the collection LμF(풞n) of all μF-subgroups of 풞n is a sublattice of L(풞n). It is also proved that LμF(풞m ×풞n) is not a sublattice of L(풞m ×풞n) in general. Further, a characterization is obtained for LμF(풞m ×풞n) to be a sublattice of L(풞m ×풞n). [ABSTRACT FROM AUTHOR]

In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups G are studied in respect of formation of lattices L(G) and sublattices of L(G). It is proved that the collections of all pronormal subgroups of An and Sn do not form sublattices of respective L(An) and L(Sn), whereas the collection of all pronormal subgroups LPrN(Dicn) of a dicyclic group is a sublattice of L(Dicn). Furthermore, it is shown that L(Dicn) and LPrN(Dicn) are lower semimodular lattices. [ABSTRACT FROM AUTHOR]

In this paper, the collections of all pronormal subgroups of Dn and Hall subgroups for groups An, Sn and Dn are studied. It is proved that the collection of all pronormal subgroups of Dn is a sublattice of L(Dn). It is also proved that the collection of all Hall subgroups of Dn, An and Sn do not form sublattices of respective L(Dn), L(An) and L(Sn). [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the collection of pronormal subgroups of dihedral groups Dn for different values of n. We enumerate the number of pronormal subgroups of Dn when n is some power of 2. Also, the relation of the collection of all pronormal subgroups with normal subgroups and all subgroups of Dn for different values of n are studied. [ABSTRACT FROM AUTHOR]

By several postulates we introduce a new class of algebraic lattices, in which a main role is played by so called normal elements. A model of these lattices are weak-congruence lattices of groups, so that normal elements correspond to normal subgroups of subgroups. We prove that in this framework many basic structural properties of groups turn out to be lattice-theoretic. Consequently, we give necessary and sufficient conditions under which a group is Hamiltonian, Dedekind, abelian, solvable, supersolvable, metabelian, finite nilpotent. These conditions are given as lattice-theoretic properties of a lattice with normal elements. [ABSTRACT FROM AUTHOR]

We present a method for determining the number of decimation classes of density δ binary vectors indexed by a finite abelian group G of size ℓ and exponent ℓ ∗ such that δ is relatively prime to ℓ ∗ . This method is the first which is not based on exhaustive vector generation and exploits the subgroup lattice of Z ℓ ∗ × . Instead, our method is based on our newly developed theory of multipliers for arbitrary subsets of finite abelian groups, our results on orbits under the action of the multiplier group, and finding the number of solutions of a potentially highly symmetric subset sum problem. Implementing our method on vectors indexed by Z ℓ of odd length ℓ and density (ℓ + 1) / 2 greatly increased the number of ℓ for which the number of decimation classes of such vectors is known. Additionally, our newly developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing. [ABSTRACT FROM AUTHOR]

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$ - central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$ -central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST -group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$ , where $\mathfrak{N}$ is the class of all nilpotent groups. [ABSTRACT FROM AUTHOR]

For a finite group G, the intersection graph of G denoted Γ (G) 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 H ∩ K ≠ 1. 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. [ABSTRACT FROM AUTHOR]

In this work, we study when an aggregation operator preserves the structure of T -subgroup of groups whose subgroup lattice is a chain. There are two widely used ways of defining the aggregation of structures in fuzzy logic, previously named on sets and on products. We will focus our attention on the one called aggregation on products. When the lattice of subgroups is not a chain, it is known that the dominance relation between the aggregation operator and the t-norm is crucial. We show that this property is again important for some of the groups in this study. However, for the rest of them, we must define a new property weaker than domination, that will allow us to characterize those operators which preserve T -subgroups. [ABSTRACT FROM AUTHOR]

In this note we are going to survey several invariants of finite groups related either to theirorders or to generating sets or to lattices of subgroups. Some relations among these invariants will be exhibited. Special attention will be paid to monotonicity of them.

In this paper we use group theoretic tools to obtain random variables which violate linear rank inequalities, that is inequalities which always hold on ranks of subspaces. We consider ten of the 24 (non-Shannon type) generators of linear rank inequalities in five variables and look at them as group inequalities. We prove that for primes p,q, groups of order pq always satisfy these ten group inequalities. We give partial results for groups of order p²q, and find that the symmetric group S4 is the smallest group to yield violations for two among the ten group inequalities. [ABSTRACT FROM AUTHOR]

We compute the number of symmetricr-colorings and the number of equivalence classes of symmetricr-colorings of the dihedral groupD3. [ABSTRACT FROM PUBLISHER]

In this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. Some relations among these invariants will be exhibited. Special attention will be paid to monotonicity of them. [ABSTRACT FROM AUTHOR]

Partially ordered groups satisfying the interpolation condition (and not necessarily directed) are considered. It is proved that an isomorphism theorem holds for these groups (this theorem fails to hold for partially ordered groups in the general case). A criterion for almost orthogonality of positive elements of interpolation groups is found. The location of a subgroup associated with a pair of almost orthogonal elements in the lattice of subgroups of an interpolation group is described. [ABSTRACT FROM AUTHOR]

Abstract: We study the lattice of subgroups of a Chevalley group over a ring A, containing its elementary subgroup over a subring . The standard description asserts that for any there exists a unique subring of A such that H contains as a normal subgroup. The standard description was obtained by Ya. Nuzhin for algebraic field extensions. On the other hand, it was expected that the standard description does not hold for transcendental extensions. This was recently proved by the author for simply laced root systems. Now, suppose that Φ is doubly laced, i.e. , or , and that 2 is invertible in S. In the article we prove that in these settings the standard description of holds for an arbitrary pair of rings . [Copyright &y& Elsevier]

Abstract: We investigate subgroups of a Chevalley group over a ring A, containing its elementary subgroup over a subring . Assume that the root system Φ is simply laced and is a polynomial ring. We show that if G is of adjoint type, then there exists an element such that , where denotes the subgroup, generated by a set X, and * stands for the free product. It follows that under the above assumptions the lattice is not standard. Moreover, combining the above result with theorems of Nuzhin and the author one obtains a necessary and sufficient condition for to be standard provided that A and F are fields of characteristic not 2 and . [ABSTRACT FROM AUTHOR]

We study the partially prefrattini groups of a finite soluble group. We prove that the set of all partially prefrattini subgroups associated with the Gaschütz system of complements to crowns is a Boolean lattice. [ABSTRACT FROM AUTHOR]

We consider description of the lattice L of subgroups between En(R) and GLn(A), where R is a subring of a commutative ring A. The standard answer to this problem is that the lattice L is divided into certain sandwiches parameterized by all rings between R and A. In this article, we show that the standard answer fails whenever there are α ∈ A and an epimorphism R[α] → K[x] taking R to a field K. A ring extension R ⊆ A is called quasi-algebraic if A contains no elements a with the above property. We investigate properties of quasi-algebraic extensions. [ABSTRACT FROM AUTHOR]

The main result of this paper is: any primary Arguesian lattice over the field GF(p) of geometric dimension at least three is isomorphic to the lattice of all submodules of a finitely generated module over the ring of polynomials of bounded degree over the field GF(p). [ABSTRACT FROM AUTHOR]

The sizes of Boolean combinations of subgroups G i of a finite abelian group depends only on the Boolean expression, the 0-1-sublattice generated by the G i, and the size of minimal subquotients from this sublattice. Moreover, they increase, monotonically, with those sizes. [ABSTRACT FROM AUTHOR]