The main focus in this work is to establish that L-group theory, which uses the language of functions instead formal set theoretic language, capable capturing most refined ideas and concepts classical group theory. We demonstrate by extending notion subnormality L-setting investigating its properties. develop a mechanism tackle join problem subnormal L-subgroups. conjugate L-subgroup as defined...

a subgroup $h$ is said to be $s$-permutable in a group $g$‎, ‎if‎ ‎$hp=ph$ holds for every sylow subgroup $p$ of $g$‎. ‎if there exists a‎ ‎subgroup $b$ of $g$ such that $hb=g$ and $h$ permutes with every‎ ‎sylow subgroup of $b$‎, ‎then $h$ is said to be $ss$-quasinormal in‎ ‎$g$‎. ‎in this paper‎, ‎we say that $h$ is a weakly $ss$-quasinormal‎ ‎subgroup of $g$‎, ‎if there is a normal subgroup ...

a subgroup $h$ is said to be $s$-permutable in a group $g$‎, ‎if‎ ‎$hp=ph$ holds for every sylow subgroup $p$ of $g$‎. ‎if there exists a‎ ‎subgroup $b$ of $g$ such that $hb=g$ and $h$ permutes with every‎ ‎sylow subgroup of $b$‎, ‎then $h$ is said to be $ss$-quasinormal in‎ ‎$g$‎. ‎in this paper‎, ‎we say that $h$ is a weakly $ss$-quasinormal‎ ‎subgroup of $g$‎, ‎if there is a normal subgroup ...

The concept of dominion (in the sense of Isbell) is investigated in several varieties of nilpotent groups. A complete description of dominions in the variety of nilpotent groups of class at most 2 is given, and used to prove nontriviality of dominions in the variety of nilpotent groups of class at most c for any c>1 . Some subvarieties of N2 , and the variety of all nilpotent groups of class at...

in this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite hamiltonian group. the coverings with three proper subgroups and the principal series of such a group are also counted. finally, we give a complete description of the lattice of characteristic subgroups of a finite h...

Let σ={σi:i∈I} be a partition of the set all prime numbers. A subgroup H finite group G is said to σ-subnormal in if can joined by chain subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 normal Hj or Hj/CoreHj(Hj−1) σi-group some i∈I. B soluble normalising Nσ-residual non-σ-subnormal G, where Nσ saturated formation σ-nilpotent groups. We show that normalises does not have section σ-residu...

Let G be a connected, simply connected nilpotent Lie group, identified with real algebraic subgroup of UT(n,R), and let Γ lattice in G, π:G→G∕Γ the quotient map. For semialgebraic X⊆G, more generally definable set an o-minimal structure on field, we consider topological closure π(X) compact nilmanifold G∕Γ. Our theorem describes cl(π(X)) terms finitely many families cosets subgroups G. The unde...

In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore,...

Let B be a p-block of a finite group, and set m = ∑ χ(1), the sum taken over all height zero characters of B. Motivated by a result of M. Isaacs characterising p-nilpotent finite groups in terms of character degrees, we show that B is nilpotent if and only if the exact power of p dividing m is equal to the p-part of |G : P ||P : R|, where P is a defect group of B and where R is the focal subgro...

The plausibility and desirability of such a functor E were shown by Adams [2]. To obtain an existence proof (§3), I will construct an appropriate localization functor on the category of simplicial sets and will show that it induces the desired h,-localization functor on Ho. The backbone of this proof is in an appendix ($10~$12), where I introduce a version of simplicial homotopy theory in which...