Basic subgroups in modular abelian group algebras

Suppose F is a perfect field of char F = p ≠ 0 and G is an arbitrary abelian multiplicative group with a p-basic subgroup B and p-component G _p . Let FG be the group algebra with normed group of all units V(FG) and its Sylow p-subgroup S(FG), and let I _p (FG; B) be the nilradical of the relative augmentation ideal I(FG; B) of FG with respect to B. The main results that motivate this article are that 1 + I _p (FG; B) is basic in S(FG), and B(1 + I _p (FG; B)) is p-basic in V(FG) provided G is p-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when G is p-primary. Thus the problem of obtaining a (p-)basic subgroup in FG is completely resolved provided that the field F is perfect. Moreover, it is shown that G _p (1 + I _p (FG; B))/G _p is basic in S(FG)/G _p , and G(1 + I _p (FG; B))/G is basic in V(FG)/G provided G is p-mixed. As consequences, S(FG) and S(FG)/G _p are both starred or divisible groups. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.     

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.     

Finite groups with <Emphasis Type="Italic">S</Emphasis>-supplemented <Emphasis Type="Italic">p</Emphasis>-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B _sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B _sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.     

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.     

Invariants for group algebras of Abelian groups

LetF be a field of characteristicp>0 and letG be an arbitrary abelian group written multiplicatively withp-basis subgroup denoted byB. The first main result of the present paper is thatB is an isomorphism invariant of theF-group algebraFG. In particular, thep-local algebraically compact groupG can be retrieved fromFG. Moreover, for the lower basis subgroupB ₁ of thep-componentG _p it is shown thatG _p/B_l is determined byFG. Besides, ifH is (p-)high inG, thenG _p/H_p andH ^{p n}[p] for ℕ₀ are structure invariants forFG, andH[p] as a valued vector space is a structural invariant forN ₀ G, whereN _p is the simple field ofp-elements. Next, presume thatG isp-mixed with maximal divisible subgroupD. ThenD andF(G/D) are functional invariants forFG. The final major result is that the relative Ulm-Kaplansky-Mackeyp-invariants ofG with respect to the subgroupC are isomorphic invariants of the pair (FG, FC) ofF-algebras. These facts generalize and extend analogous in this aspect results due to May (1969), Berman-Mollov (1969) and Beers-Richman-Walker (1983). As a finish, some other invariants for commutative group algebras are obtained.     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

Embeddings into Orthomodular Lattices with Given Centers,State Spaces and Automorphism Groups

We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space affinely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its subalgebra.     

On Brauer’s <Emphasis Type="Italic">k</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)-problem for blocks with metacyclic defect groups of odd order

Let G be a finite group, and suppose that B is a p-block of G with defect group D. Let k(B) denote the number of ordinary irreducible characters in B. It was conjectured by Brauer that k(B) ≤ |D|. In this paper, we will prove Brauer’s inequality in the case that D is metacyclic and p is odd.     

On SS-Quasinormal Subgroups of Finite Groups

A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.     

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 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.     

Finite groups with S-quasinormally embedded or SS-quasinormal subgroups

Suppose that G is a finite group and H is a subgroup of G. H 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; H is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G=HB and H permutes with every Sylow subgroup of B. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1<|D|<|P| and study the structure of G under the assumption that every subgroup H of P with |H|=|D| is either S-quasinormally embedded or SS-quasinormal in G. Some recent results are generalized and unified.     

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.

     

Essential independent sets and Hamiltonian cycles

An independent set S of a graph G is said to be essential if S has a pair of vertices distance two apart in G. We prove that if every essential independent set S of order k ≥ 2 in a k-connected graph of order p satisfies max {deg v:v ϵ S} ≥ ½ p, then g is hamiltonian. This generalizes the result of Fan (J. Combinatorial Theory B 37 (1984), 221–227). If we consider the essential independent sets of order k + 1 instead of k in the assumption of the above statement, we can no longer assure the existence a hamiltonian cycle. However, we can still give a lower bound to the length of a longest cycle. © 1996 John Wiley & Sons, Inc.     

The rectifiability threshold in abelian groups

For any abelian group G and integer t ≥ 2 we determine precisely the smallest possible size of a non-t-rectifiable subset of G. Specifically, assuming that G is not torsion-free, denote by p the smallest order of a non-zero element of G. We show that if a subset S ⊆ G satisfies |S| ≤ ⌌log _t p⌍, then S is t-isomorphic (in the sense of Freiman) to a set of integers; on the other hand, we present an example of a subset S ⊆ G with |S| = ⌌log _t p⌍ + 1 which is not t-isomorphic to a set of integers.     

A tensor product of the Steinberg module and a certain simple kG(pr)-module

Let G be a connected, semisimple, and simply connected algebraic group defined and split over the finite field of order p, and let G(q) be the corresponding finite Chevalley or twisted group, where q = p^r. Recently, Anwar determines the direct sum decomposition of the tensor product of the rth Steinberg module and a simple G-module with a (p,r)-minuscule highest weight λ. In this paper, we determine that of the tensor product regarded as a module for G(q) under some weak assumptions for λ.     

Barsotti–Tate groups and p-adic representations of the fundamental group scheme

On a scheme S over a base scheme B we study the category of locally constant BT groups, i.e. groups over S that are twists, in the flat topology, of BT groups defined over B. These groups generalize p-adic local systems and can be interpreted as integral p-adic representations of the fundamental group scheme of S/B (classifying finite flat torsors on the base scheme) when such a group exists. We generalize to these coefficients the Katz correspondence for p-adic local systems and show that they are closely related to the maximal nilpotent quotient of the fundamental group scheme.     

A module structure on cyclic cohomology of group graded algebras

Letk be a field of characteristic 0, and letB be an algebra overk which is graded by a discrete groupG. Let HC*(A) denote the cyclic cohomology of an algebraA overk. We prove that there is an HC*(kG)-module structure on HC*(B) which generalizes Connes' periodicity operator on HC*(B). This module structure also decomposes with respect to conjugacy classes and results in a natural generalization of the results of Burghelea and Nistor in the cases of group algebras and algebraic crossed product algebras, respectively. Moreover, the proofs given in this paper are purely analytic with explicit constructions which can be used in the calculation of the cyclic cohomology of topological twisted crossed product algebras.Research sponsored in part by NSF Grant DMS-9204005.     

Restricted sumsets and a conjecture of Lev

LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev. This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.