Narrow Results

Let G be a finite group and p an odd prime. Let be the set of proper subgroups M of G with |G:M| not a prime power and |G:M|_p=1. In this paper, we investigate the structure of G if every member of is nilpotent. In particular, a new characterization of PSL(2,7) is obtained. The proof of the theorem depends on the classification of finite simple groups.

Let bcl(G) denote the largest conjugacy class length of a finite group G. In this note, we prove that if bcl(G)<p² for a prime p, then |G:O_p(G)|_p≤p.

It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.

Let G be a finite group. We say that a subgroup H of G is -normal in G if G has a subnormal subgroup T such that TH = G and (H ∩ T)H_G/H_G is contained in the -hypercenter of G/H_G, where is the class of the finite supersoluble groups. We study the structure of G under the assumption that some subgroups of G are -normal in G.

It is proved that the lattice of totally saturated formations of finite groups is distributive. Thus, we give an affirmative answer to the problem proposed by Shemetkov, Skiba and Guo.

In this paper, we prove that for any prime p, a finite group with 18p elements having maximal order is solvable. This result provides some information in solving an open problem proposed by Thompson.

Let H and T be subgroups of a finite group G. We say that H is completely c-permutable with T in G if there exists an element x ∈ 〈H,T〉 such that HT^x = T^xH. In this paper, we use this concept to determine the supersolubility of a group G = AB, where A and B are supersoluble subgroups of G. Some criterions of supersolubility of such groups are obtained and some known results are generalized.

Let G be a finite group. By using the concept of strong θ-completions for maximal subgroups of G, we obtain in this paper some new conditions under which G is solvable or supersolvable.

Let G be a finite group and p a prime. We say that G is quasisupersoluble (resp., p-quasisupersoluble) if for every non-cyclic chief factor H/K of G (resp., for every non-cyclic chief factor H/K of G of order divisible by p), every automorphism of H/K induced by an element of G is inner. In this paper, we study the structure of quasisupersoluble and p-quasisupersoluble finite groups.

Let ω be a set of primes with |ω| > 1, and m > n ≥ 0 be integers. It is proved that the lattice of all τ-closed m-multiply ω-saturated formations is not a sublattice of the lattice of all τ-closed n-multiply ω-saturated formations.

In [7] we have introduced the Frobenius categories over a finite p-group P, and we have associated to — suitably endowed with some central k*-extensions — a “Grothendieck group” as an inverse limit of Grothendieck groups of categories of modules in characteristic p obtained from , determining its rank. Our purpose here is to introduce an analogous inverse limit of Grothendieck groups of categories of modules in characteristic zero obtained from , determining its rank and proving that its extension to a field is canonically isomorphic to the direct sum of the corresponding extensions of the “Grothendieck groups” above associated with the centralizers in of a suitable set of representatives of the -classes of elements of P.

For a non-abelian group G, we associate the non-commuting graph ∇ (G) whose vertex set is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this paper, we prove that Aut(J₂) and Aut(M^cL) are characterized by their non-commuting graphs.

The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups A_p+3 are OD-characterizable, while the symmetric groups S_p+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut(O₇(3)) and Aut(S₆(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

In this paper, we mainly consider the relationship between the complex irreducible characters of a π-separable group and the complex irreducible characters of its Hall π-subgroup. If a π-group S acts on a π-separable group G, let H be an S-invariant Hall π-subgroup of G and C_NG(H)/H(S)=1. Then we construct a natural bijection from the set Lin_S(H) onto the set Irr_π′,S(G). Furthermore, we get a bijection from the linear characters of H onto Irr_π′(G).

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of isomorphism classes of simple kG-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.

Let G be a finite group with a normal Sylow p-subgroup P. Let be a complete discrete valuation ring with residue field F of characteristic p. Let M be an indecomposable endo-monomial -module. In this paper we prove that M extends to an -module if and only if M is G-stable. A similar and well-known version for endo-permutation modules is due to Dade.

A finite simple group G is called a simple K₃-group if |G| has exactly three distinct prime divisors. In this paper, we characterize the finite simple K₃-groups and some L₂(p) only by using the group order and maximal element order, which generalize some known results.

Let be the class of groups of non-prime-power order or the class of groups of prime-power order. In this paper we give a complete classification of finite non-solvable groups with a quite small number of conjugacy classes of -subgroups or classes of -subgroups of the same order.

We introduce a new avatar of a Frobenius P-category under the form of a suitable subring of the double Burnside ring of P — called the Hecke algebra of — where we are able to formulate: (i) the generalization to a Frobenius P-category of the Alperin Fusion Theorem, (ii) the “canonical decomposition” of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and (iii) the “basic P × P-sets” in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P × P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.