In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal embeddings of finite groups into finite groups with additional properties we bring some illustration to a result of H. Heineken and J. Lennox.
In 1974, Helmut Wielandt proved that in a finite group $G$, a subgroup $A$ is subnormal if and only if it is subnormal in every $\seq{A,g}$ for all $g\in G$. In this paper, we prove that the subnormality of an odd order nilpotent subgroup $A$ of $G$ is already guaranteed by a seemingly weaker condition: $A$ is subnormal in $G$ if for every conjugacy class $C$ of $G$ there exists $c\in C$ for which $A$ is subnormal in $\seq{A,c}$. We also prove the following property of finite non-abelian simple groups: if $A$ is a subgroup of odd prime order $p$ in a finite almost simple group $G$, then there …
Let $G$ be a finite group and let $\mathfrak{F}$ be a hereditary saturated formation. We denote by $\mathbf{Z}_{\mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $\mathfrak{F}$-central in $G$, that is, \[ (H/K) \rtimes (G/\mathbf{C}_{G}(H/K)) \in \mathfrak{F}. \]A subgroup $A \leq G$ is said to be $\mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$\mathfrak{F}$-subnormal in $G$, if there is a subgroup chain \[ A = A_0 \leq A_1 \leq \ldots \leq A_n = G \] such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_i / (A_{i-1…
Let $\frak {F}$ be a class of group. A subgroup $A$ of a finite group $G$ is said to be $K$-$\mathfrak{F}$-subnormal in $G$ if there is a subgroup chain $$A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$$ such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{F}$ for all $i=1, \ldots , n$. A formation $\frak {F}$ is said to be $K$-lattice provided in every finite group $G$ the set of all its $K$-$\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$. In this paper we consider some new applications of the theory of $K$-lattic…
In this article, we study group theoretical embedding properties of subgroups in central products of finite groups. Specifically, we give characterizations of normal, subnormal, and abnormal subgroups of a central product of two groups.
We investigate the intersection of normalizers and $\mathfrak{F}$-subnormalizers of different types of systems of subgroups ($\mathfrak{F}$-maximal, Sylow, cyclic primary). We described all formations $\mathfrak{F}=\underset{i\in I}\times\mathfrak{F}_{π_i}$ for which the intersection of normalizers of all $\mathfrak{F}_i$-maximal subgroups of $G$ is the $\mathfrak{F}$-hypercenter of $G$ for every group $G$. Also we described all formations $\mathfrak{F}$ for which the intersection of $\mathfrak{F}$-subnormalizers of all Sylow (cyclic primary) subgroups of $G$ is the $\mathfrak{F}$-hypercenter …
We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the sense that it cannot be chosen to be neither solvable of length 2, nor a nilpotent group. For any non-trivial word set $V \subseteq F_\infty$ there is an order-preserving verbal embedding of $\mathbb{Q}$ into a 2-generator group $G$. The embeddings constructed are subnormal.
We show that every countable group H with solvable word problem (=computable group) can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G.
We study the class of finite groups $G$ satisfying $Φ(G/N)= Φ(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
Let $ σ$ be a partition of the set of all primes and $\mathfrak{F}$ be a hereditary formation. We described all formations $\mathfrak{F}$ for which the $\mathfrak{F}$-hypercenter and the intersection of weak $K$-$\mathfrak{F}$-subnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all $σ$-nilpotent groups has this property. With the help of our results we solve a particular case of L.A.~Shemetkov's problem about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter. As corollaries we obtained P. Hall's and R. Baer's class…
We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic groups.
Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.
We give a simple construction involving partial actions which permits us to obtain an easy proof of a weakened version of L. O'Carroll's theorem on idempotent pure extensions of inverse semigroups.
We show that if $G$ is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group $G^*$, such that $G$ is a subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.
We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.
A Magnus subgroup of a one-relator group is the free subgroup freely generated by a proper subset of the generators. Two such subgroups can intersect in the obvious way or in a larger, exceptional way. The condition of non-exceptional intersection of Magnus subgroups of a one-relator group is applied to give criteria for the resulting cyclically presented groups to contain nonabelian free subgroups.
Using suitable deformations of simplicial trees and the duality theory for median sets, we show that every free action on a median set can be extended to a free and transitive one. We also prove that the category of median groups is a reflective full subcategory of the category of free actions on pointed median sets.
The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-Hölder theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantu…
We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension $2$, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension $2$, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.
Hahn's embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn's theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial sublex product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess o…
