The signatures of the congruence subgroups G 0(τ) of the hecke groups G 4 and G 6

We determine the signatures of the congruence subgroups of the Hecke groups G ₄ and G ₆.     

Finite Groups with Few TI-Subgroups

A subgroup A of a finite group G is called a TI-subgroup if either A ∩ A ^x  = 1 or A ∩ A ^x  = A holds for all x ∈ G. In this paper, finite group all of whose meta-cyclic subgroups are TI-subgroups are classified completely. In particular, such groups are solvable.     

On the Size of Boolean Combinations of Subgroups of Finite Abelian Groups

The sizes of Boolean combinations of subgroups G _i of a finite abelian group depends only on the Boolean expression, the 0-1-sublattice generated by the G _i , and the size of minimal subquotients from this sublattice. Moreover, they increase, monotonically, with those sizes.     

Groups with π-maximal and π-layer maximal conditions

By analogy with π-minimal, π-layer minimal, primary minimal, and layer minimal conditions, similar maximal conditions are introduced and studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 311–320, August, 2000.     

Injectors and Radicals in Products of Totally Permutable Groups

Abstract

Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied.     

Parametrization of tamely ramified maximal tori using bounded subgroups

Let be a reductive group defined over a local complete field F with discrete valuation, and split over some unramified extension of F, and let G be its group of F-points. In this paper, we define a class of abelian “torus-like” subgroups in nonreductive groups, called pseudo-tori, which generalizes the notion of torus, and we establish a correspondence between conjugacy classes of tamely ramified maximal tori of G and association classes of maximal pseudo-tori of the quotients of parahorics of G by their second congruence subgroup, viewed as groups of k-points of algebraic groups defined over the residual field k of F.      

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.     

The influence of <Emphasis Type="Italic">X</Emphasis>-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771172, 10771180)     

Finitely Presented Wreath Products and Double Coset Decompositions

We characterize which permutational wreath products $G \ltimes W^{(X)}We characterize which permutational wreath products are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X ². On the one hand, this extends a result of G. Baumslag about infinite presentation of standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.      

A note on the conjugacy classes of non-cyclic subgroups

Let G be a finite group and δ(G) denote the number of conjugacy classes of all non-cyclic subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In [7 Meng, W., Li, S. R. (2014). Finite groups with few conjugacy classes of non-cyclic subgroups. Scientia Sin. Math. 44:939–944.[Crossref] , [Google Scholar]], Meng and Li showed the inequality δ(G)≥2^|π(G)|?2, where G is non-cyclic solvable group. In this paper, we describe the finite groups G such that δ(G) = 2^|π(G)|?2. Another aim of this paper would show δ(G)≥M(G)+2 for unsolvable groups G and the equality holds ?G?A₅ or SL(2,5), where M(G) denotes the number of conjugacy classes of all maximal subgroups of G.