A group G is simple if and only if the diagonal subgroup of G ×G is a maximal subgroup. This striking property is very easy to prove and raises the question of determining all the maximal subgroups of G , where G denotes the direct product of n copies of G . The first purpose of this note is to answer completely this question. We show in particular that if G is perfect, then any maximal subgrou...

A theorem of O’Nan and Scott [6]; [2, Chapter 4] restricts the possibilities for maximal subgroups of finite symmetric groups: they are of six types of which the first four are explicitly known, the fifth involves a finite simple group, and the sixth an action of a simple group. This result, in conjunction with the Classification of Finite Simple Groups, has a number of consequences. In particu...

We prove that if a countable group is elementarily equivalent to non-abelian free and all of its abelian subgroups are cyclic, then the union chain regular NTQ groups (i.e., hyperbolic towers).

The main goal of this paper is to compute the maximal abelian dimension of each solvable nondecomposable Lie algebra of dimension less than 7. To do it, we apply an algorithmic method which goes ruling out non-valid maximal abelian dimensions until obtaining its exact value. Based on Mubarakzyanov and Turkowsky’s classical classifications of solvable Lie algebras (see [13] and [19]) and the cla...