Let G be a finite group with a non-abelian minimal normal subgroup N which is a direct product of the simple group X. The maximal subgroups of G which complement N and their conjugacy classes are parametrised in terms of certain homomorphisms taking values in AutX and satisfying particular conditions.

We build two nonabelian CSA-groups in which maximal abelian subgroups are conjugate and divisible, as the countable unions of increasing chains of CSA-groups and by keeping the constructions as free as possible in each case. For n ≥ 1, let Fn denote the free group on n generators. We view all groups G as first-order structures 〈G, ·, , 1〉, where ·, , and 1 denote respectively the multiplication...

Let $\mathbf{K}$ be an algebraically closed field. The Cremona group $\operatorname{Cr}_2(\mathbf{K})$ is the of birational transformations projective plane $\mathbb{P}^2_{\mathbf{K}}$. We carry out overall study centralizers elements infinite order in which leads to a classification embeddings $\mathbf{Z}^2$ into $\operatorname{Cr}_2(\mathbf{K})$, as well maximal non-torsion abelian subgroups ...

Let G be a finite group with an abelian normal subgroup N . When does N have a unique conjugacy class of complements in G? We consider this question with a focus on subgroups and properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup (Theorem 1 of [3]). Furthermore, w...

We show that the set SA(G) of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group G is dense in the product of the sets SA(P) over all maximal parabolic subgroups P. The set BSA(G) of equivalence classes of biautomatic structures on G is isomorphic to the product of the sets BSA(P) over the cusps (conjugacy classes of maximal parabolic subgroups)...

In this paper, without using the classification of finite simple groups, we determine the structure of  finite simple groups whose Sylow 3-subgroups are of the order 9. More precisely, we classify finite simple groups whose Sylow 3-subgroups are elementary abelian of order 9.

let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

In this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. It is proved that an abelian group of finite exponent is strongly coatomic.