The Cohomology of the Sylow 2-Subgroup of J2

The Hall-Janko-Wales group J₂ is one of the twenty-six sporadicfinite simple groups. The cohomology of its Sylow 2-subgroupS_J is computed, an important step in calculating the mod 2 cohomologyof J₂. The spectral sequence corresponding to the central extensionfor S_J is described and shown to collapse at the eighth page.The group S_J contains two subgroups (the central product of a dihedral and a quaternionic group)and 2²⁺⁴ (the Sylow 2-subgroup of the matrix group PSL₃(₄))which detect the cohomology of S_J. The cohomology relationsfor the subgroup 2²⁺⁴ are computed.     

Finite Group Actions with Fixed Points on Homology 3-Spheres

 If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds. Received 25 September 1997; in revised form 2 June 1998     

Relative Group Cohomology and The Orbit Category

Let G be a finite group and ? be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ?-projective resolution for ? when ? is the family of all subgroups H ≤ G with rk H ≤ rkG ? 1. We answer this question negatively by calculating the relative group cohomology ?H*(G, 𝔽₂) where G = ?/2 × ?/2 and ? is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ?H*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ?. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E ₂ page is isomorphic to the relative group cohomology of G.     

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.     

Maximal 2-Local Geometries for the Symmetric Groups

Abstract

The set 𝒩_max (G, T) consisting of all maximal 2-local subgroups of G = Sym(n) which contain T, a Sylow 2-subgroup of G, is investigated. In addition to determining the structure of the subgroups in 𝒩_max (G, T), the simplicial sets of maximal rank are classified.     

On Nearly S-Permutable Subgroups of Finite Groups

We say that a subgroup H of a finite group G is nearly S-permutable in G if for every prime p such that (p, |H|) = 1 and for every subgroup K of G containing H the normalizer N _K (H) contains some Sylow p-subgroup of K. We study the structure of G under the assumption that some subgroups of G are nearly S-permutable in G.     

Bounding an Index by the Largest Character Degree of a p-Solvable Group

In this article, we show that if p is a prime and G is a p-solvable group, then |G: O _p (G)| _p  ≤ (b(G) ^p /p)^1/(p?1), where b(G) is the largest character degree of G. If p is an odd prime that is not a Mersenne prime or if the nilpotence class of a Sylow p-subgroup of G is at most p, then |G: O _p (G)| _p  ≤ b(G).     

Indecomposable Sylow 2-Subgroups of Simple Groups

Let S be a Sylow 2-subgroup of a finite simple group and let S=S₁×S₂××S_k be the direct product and each component S_i, i=1,2,...,k is indecomposable. In this article, we prove that each S_i is also a Sylow 2-subgroup of some simple group. Mathematics Subject Classifications (2000) 20E32, 20D20.     

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.     

On Weakly s-permutably Embedded Subgroups of Finite Groups

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutably 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-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup H _se of G contained in H such that G = HT and H ∩ T ≤ H _se . We investigate the influence of weakly s-permutably embedded subgroups on the structure of finite groups. Some recent results are generalized.     

On the Artin exponent of some rational groups

A finite group G is called rational if all its irreducible complex characters are rational valued. In this paper, we show that if G is a direct product of finitely many rational Frobenius groups then every rationally represented character of G is a generalized permutation character. Also we show that the same assertion holds when G is a solvable rational group with a Sylow 2-subgroup isomorphic to the dihedral group of order 8 and an abelian normal Sylow 3-subgroup.     

On 𝔬𝔰𝔭(1 | 2)-Relative Cohomology on S 1|1

We compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the module of λ-densities 𝔉_λ on S ^1|1. This result allows us to compute the second 𝔬𝔰𝔭(1 | 2) ?relative cohomology space of 𝒦(1) with coefficients in the Poisson superalgebra 𝒮𝒫. We explicitly give 2-cocycles spanning these cohomology spaces.     

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.     

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.     

A note on 2-nilpotence of finite groups

Ballester-Bolinches and Guo showed that a finite group G is 2-nilpotent if G satisfies: (1) a Sylow 2-subgroup P of G is quaternion-free and (2) Ω₁(P ∩  G′)?≤?Z(P) and N _G (P) is 2-nilpotent. In this paper, it is obtained that G is a non-2-nilpotent group of order 16q for an odd prime q satisfying (1) a Sylow 2-subgroup P of G is not quaternion-free and (2) Ω₁(P?∩?G′)?≤?Z(P) and N _G (P) is 2-nilpotent if and only if q?=?3 and G???GL ₂(3).     

The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical

Let G be a finite group, and let p ₁,…, p _m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P ₁,…, P _m , where P _i is a Sylow p _i -subgroup of G, and g ∈ G, denote by m _𝒫(g) the number of factorizations g = g ₁…g _m such that g _i  ∈ P _i . Previously, it was shown that the properly normalized average of m _𝒫 over all 𝒫 is a complex character over G termed the Average Sylow Multiplicity Character. The present article identifies the kernel of this character as the subgroup of G consisting of all g ∈ G such that m _𝒫(gx) = m _𝒫(x) for all 𝒫 and all x ∈ G. This result implies a close connection between the kernel and the solvable radical of G.     

Local c*-Supplementation of Some Subgroups in Finite Groups

A subgroup H of a finite group G is called c*-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormally embedded in G. In this paper, we investigate the local c*-supplementation of maximal subgroups of some Sylow p-subgroup and present some sufficient and necessary conditions for a finite group to be p-nilpotent. As applications, we give some sufficient conditions for a finite group to be in a saturated formation.     

A remark on Gelfand–Graev characters for finite simple groups

The famous Gelfand–Graev character of a group of Lie type G is a multiplicity free character of shape ν ^G , where ν is a suitable degree 1 character of a Sylow p-subgroup and p is the defining characteristic of G. We show that, for an arbitrary non-abelian simple group G, if ν is a linear character of a Sylow p-subgroup of G such that ν ^G is multiplicity free, then G is isomorphic to either a group of Lie type in defining characteristic p, or to a group PSL(2, q), where either p = q + 1, or p = 2 and q + 1 or q ? 1 is a 2-power.     

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.     

Polynomial-time algorithms for finding elements of prime order and sylow subgroups

Assume that generators are given for a subgroup G of the symmetric group S_n of degree n, and that r is a prime dividing |;G|;. Polynomial-time algorithms are given for finding an element of G of order r, and for finding a Sylow r-subgroup of G if G is simple.