Blocks of full defect with nonabelian metacyclic defect groups

Let p be an odd prime and let B be a p-block of a finite group G with a nonabelian metacyclic defect group P which is a Sylow p-subgroup of G. The purpose of this article is to study the ordinary and modular irreducible characters in B. In particular, we calculate k _i (B) and l _i (B) for an arbitrary nonnegative integer i.     

On two classes of finite supersoluble groups

Let ? be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ?-S-semipermutable if H permutes with every Sylow p-subgroup of G in ? for all p?π(H); H is said to be ?-S-seminormal if it is normalized by every Sylow p-subgroup of G in ? for all p?π(H). The main aim of this paper is to characterize the ?-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ? are ?-S-semipermutable in G and the ?-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ? are ?-S-seminormal in G.     

Unfused Involutions in Finite Groups

ABSTRACT

Let x be a p-element of a finite group G. We say that x is unfused in G if, for some Sylow p-subgroup S of G containing x, all G-conjugates of x in S are S-conjugates. It is shown (using the classification of finite simple groups) that a finite group that contains an unfused involution has a chief factor of order 2.

     

On the Loewy length of the center of a block with elementary abelian defect groups

Let G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.     

On permutable subgroups of finite groups

Let \frak Z \frak Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, \frak Z \frak Z contains exactly one and only one Sylow p-subgroup of G. A subgroup H of a finite group G is said to be \frak Z \frak Z -permutable if H permutes with every member of \frak Z \frak Z . The purpose here is to study the influence of \frak Z \frak Z -permutability of some subgroups on the structure of finite groups. Some recent results are generalized.     

A p-nilpotency criterion

Let G be a finite group, p a fixed prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, then G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p = 2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.     

On p-nilpotency of finite groups with some subgroups π-quasinormally embedded

Summary A subgroup H of a group G is said to be π-quasinormal in G if it permutes with every Sylow subgroup of G, and H is said to be π-quasinormally embedded in G if for each prime dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some π-quasinormal subgroups of G. We characterize p-nilpotentcy of finite groups with the assumption that some maximal subgroups, 2-maximal subgroups, minimal subgroups and 2-minimal subgroups are π-quasinormally embedded, respectively.     

On finite groups with prime-power order S-quasinormally embedded subgroups

A subgroup H of a finite group G is said to be S-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. In this paper we investigate the structure of finite groups that have some S-quasinormally embedded subgroups of prime-power order, and new criteria for p-nilpotency are obtained.     

ZEROS OF CHARACTERS ON PRIME ORDER ELEMENTS

Suppose that G is a finite group, let χ be a faithful irreducible character of degree a power of p and let P be a Sylow p-subgroup of G. If χ(x) ≠ 0 for all elements of G of order p, then P is cyclic or generalized quaternion.

     

On finite groups for which the lattice of <Emphasis Type="Italic">S</Emphasis>-permutable subgroups is distributive

Let p be a prime and let P be a Sylow p-subgroup of a finite nonabelian group G. Let bcl(G) be the size of the largest conjugacy classes of the group G. We show that if p is an odd prime but not a Mersenne prime or if P does not involve a section isomorphic to the wreath product \({Z_p \wr Z_p}\), then \({|P/O_p(G)| \leq bcl(G)}\).     

Weakly s-semipermutable subgroups of finite groups

In this paper, we introduce the concept of weakly s-semipermutable subgroups. Let G be a finite group. Using the condition that the minimal subgroups or subgroups of order p ² of a given Sylow p-subgroup of G are weakly s-semipermutable in G, we give a criterion for p-nilpotency of G and get some results about formation.     

A Remark on З-Permutability of Finite Groups

Let З be a complete set of Sylow subgroups of a finite group G, that is, З contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to be З-permutable if it permutes with every member of З. Recently, using the Classification of Finite Simple Groups, Heliel, Li and Li proved tile following result： If the cyclic subgroups of prime order or order 4 iif p = 2） of every member of З are З-permutable subgroups in G, then G is supersolvable. In this paper, we give an elementary proof of this theorem and generalize it in terms of formation.     

Finite Groups with Some S-quasinormally Embedded Subgroups

Suppose that G is a finite group and H is a subgroup of G. H is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G; H is said to be S-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. We investigate the influence of S-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.     

An upper bound for the representation dimension of group algebras with elementary abelian Sylow p-subgroups

Abstract

In this article, we present an upper bound on the representation dimension of the group algebra of a group with an elementary abelian Sylow p-subgroup. Specifically, if k is a field of characteristic p and G is a group with elementary abelian Sylow p-subgroup P, we prove that the representation dimension of kG is bounded above by the order of P. Key to proving this theorem is the separable equivalence between the two algebras and some nice properties of Mackey decomposition.     

On p-Regular G-Conjugacy Classes and the p-Structure of Normal Subgroups

Let N be a p-solvable normal subgroup of a group G such that N contains a noncentral Sylow r (≠ p)-subgroup R of G. It is proved that the p-complements of N are nilpotent if |x ^G |=1 or m for every p-regular element x of N whose order is divisible by at most two distinct primes. Our result, therefore, gives some information concerning the nilpotence of some kind of subgroups of a group G.     

Nonsoluble length of finite groups with restrictions on Sylow subgroups

By 𝔛(n) we denote the variety of all groups satisfying the law [x,y]ⁿ≡1, that is, groups with commutators of order dividing n. Let p be a prime and G a finite group whose Sylow p-subgroups have normal series of length k all of whose quotients belong to 𝔛(n). We show that the non-p-soluble length λ_p(G) of G is bounded in terms of k and n only (Theorem 1.2). In the case where p is odd, a stronger result is obtained (Theorem 1.3).     

On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if N _G (H) ∩ H ^x  ≤ H for all x ∈ G. A subgroup H of G is called weakly ?-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ?-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ?-subgroups in G. Some recent results are extended and generalized.     

A solvability criterion for finite groups

We show that if for every prime p, the normalizer of a Sylow p-subgroup of a finite group G admits a p-solvable supplement, then G is solvable. This generalizes a solvability criterion of Hall which asserts that a finite group G is solvable if and only if G has a Hall p′-subgroup for every prime p.     

On s-Permutable Subgroups and p-Nilpotency of Finite Groups

A subgroup H of a finite group G is said to be s-permutable in G if H permutes with every Sylow subgroup of G. In this article, some sufficient conditions for a finite group G to be p-nilpotent are given whenever all subgroups with order p ^m of a Sylow p-subgroup of G are s-permutable for a given positive integer m.     

The influence of weakly-supplemented subgroups on the structure of finite groups

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In the paper it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of P∩G′ is weakly-supplemented in N _G (P), where P is a Sylow p-subgroup of G. As applications, some interesting results with weakly-supplemented minimal subgroups of P∩G′ are obtained.