We examine the structure of compact Lie groups using a finite maximal abelian subgroup A in place of a maximal torus. Just as the classical notion of roots exhibits many interesting subgroups, so the notion of roots of A exhibits many (rather different) interesting subgroups.

‎Here‎, ‎we investigate a conjecture posed by Amiri and Ariannejad claiming‎ ‎that if every maximal subfield of a division ring $D$ has trivial normalizer‎, ‎then $D$ is commutative‎. ‎Using Amitsur classification of‎ ‎finite subgroups of division rings‎, ‎it is essentially shown that if‎ ‎$D$ is finite dimensional over its center then it contains a maximal‎ ‎subfield with non-trivial normalize...

The maximality of Abelian subgroups play a role in various parts of group theory. For example, Mycielski [8, 7] has extended a classical result of Lie groups and shown that a maximal Abelian subgroup of a compact connected group is connected and, furthermore, all the maximal Abelian subgroups are conjugate. For finite symmetric groups the question of the size of maximal Abelian subgroups has be...

‎In this paper we prove that a finite group $G$ having at most three‎ ‎conjugacy classes of non-normal non-abelian proper subgroups is‎ ‎always solvable except for $Gcong{rm{A_5}}$‎, ‎which extends Theorem 3.3‎ ‎in [Some sufficient conditions on the number of‎ ‎non-abelian subgroups of a finite group to be solvable‎, ‎Acta Math‎. ‎Sinica (English Series) 27 (2011) 891--896.]‎. ‎Moreover‎, ‎we s...

This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup the general skew linear group $\GL_n(D)$ over a division ring $D$. It turns out that in case where $D$ is non-commutative, if such exist, then either it abelian or $[D:F]<\infty$. Also, $F$ infinite field $n\geq 5$, every normal $\GL_n(F)$ abelian.

The notions of nearly-maximal and near Frattini subgroups considered by J.B. Riles in [20] and the natural related notions are characterized for abelian groups.

let $g$ be a finite group and let $gk(g)$ be the prime graph of $g$. we assume that $ngeqslant 5 $ is an odd number. in this paper, we show that the simple groups $b_n(3)$ and $c_n(3)$ are 2-recognizable by their prime graphs. as consequences of the result, the characterizability of the groups $b_n(3)$ and $c_n(3)$ by their spectra and by the set of orders of maximal abelian subgroups are obtai...

We show that soluble groups G of type Bredon-FP∞ with respect to the family of all virtually cyclic subgroups of G are always virtually cyclic. In such a group centralizers of elements are of type FP∞. We show that this implies the group is polycyclic. Another important ingredient of the proof is that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic ...

We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R n-trees. We first prove that Sela's limit groups do have a free action on an R n-tree. We then prove that a finitely generated group having a free action on an R n-tree can be obtained from free abelian groups and surface groups by a finite sequence of free products and amalgamations over cyclic ...