Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

A New Criterion for Finite Noncyclic Groups

Let H be a subgroup of a group G. We say that H satisfies the power condition with respect to G, or H is a power subgroup of G, if there exists a non-negative integer m such that H = G ^m  = 〈 g ^m |g ? G 〉. In this note, the following theorem is proved: Let G be a group and k the number of nonpower subgroups of G. Then (1) k = 0 if and only if G is a cyclic group (theorem of F. Szász); (2) 0 < k < ∞ if and only if G is a finite noncyclic group; (3) k = ∞ if and only if G is a infinte noncyclic group. Thus we get a new criterion for the finite noncyclic groups.     

OD-Characterization of Some Projective Special Linear Groups Over the Binary Field and Their Automorphism Groups

The Gruenberg–Kegel graph GK(G) = (V _G , E _G ) of a finite group G is a simple graph with vertex set V _G  = π(G), the set of all primes dividing the order of G, and such that two distinct vertices p and q are joined by an edge, {p, q} ∈ E _G , if G contains an element of order pq. The degree deg _G (p) of a vertex p ∈ V _G is the number of edges incident to p. In the case when π(G) = {p ₁, p ₂,…, p _h } with p ₁ < p ₂ < … <p _h , we consider the h-tuple D(G) = (deg _G (p ₁), deg _G (p ₂),…, deg _G (p _h )), which is called the degree pattern of G. The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying condition (|H|, D(H)) = (|G|, D(G)). Especially, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we prove that the simple groups L ₁₀(2) and L ₁₁(2) are OD-characterizable. It is also shown that automorphism groups Aut(L _p (2)) and Aut(L _p+1(2)), where 2 ^p  ? 1 is a Mersenne prime, are OD-characterizable. Finally, a list of finite (simple) groups which are presently known to be k-fold OD-characterizable, for certain values of k, is presented.     

On B(n,k) 2-groups

A finite group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28.     

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.     

On Derived Equivalences for Selfinjective Algebras

We show that if A is a representation-finite selfinjective Artin algebra, then every P ^? ? K ^b(𝒫 _A ) with Hom _K(Mod?A)(P ^?,P ^?[i]) = 0 for i ≠ 0 and add(P ^?) = add(νP ^?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B ₀, B ₁,…, B _m  = B such that, for any 0 ≤ i < m, B _i+1 is the endomorphism algebra of a tilting complex for B _i of length ≤ 1.     

On B(4, 13) 2-Groups

A group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j  | 1 ≤ i, j ≤ n}| ≤k. In this article, we give a complete characterization of B(4, 13) 2-groups, and then obtain a complete characterization of B(4, 13) groups.     

Nearly s-Normality of Groups and Its Properties

We call a subgroup H of a group G nearly s-normal in G if there exists N ? G such that HN ? G and H ∩ N ≤ H _sG , where H _sG is the largest s-permutable subgroup of G contained in H. In this article, we obtain some results about the nearly s-normal subgroups and use them to characterize the structure of finite groups.     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

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).     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

The Influence of Weakly s-Supplemented Subgroups on the Structure of Finite Groups

Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _{s G} , where H _{s G} is the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this article, we investigate the structure of G under the assumption that some families of subgroups of G are weakly s-supplemented in G. Some recent results are generalized.     

Finite Groups Whose Minimal Subgroups are c-Supplemented

Let G be a finite group. A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H _G , where H _G  = Core _G (H) is the largest normal subgroup of G contained in H. In this article, we investigate the structure of a finite group G under the assumption that subgroups of prime order are c-supplemented in G. Moreover, we analyze the structure of a group G when the minimal subgroups of the generalized Fitting subgroup F?(G) of G are c-supplemented in G through the theory of formations.     

Abelian Group Gradings on Full Matrix Rings

For positive integers ? and n, several authors studied ?_?-gradings of the full matrix ring M _n (k) over a field k. In this article, we show that every (G × H)-grading of M _n (k) can be constructed by a pair of compatible G-grading and H-grading of M _n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M _n (k).     

Finite Groups all of Whose Second Maximal Subgroups are QTI-Subgroups

A subgroup H of a finite group G is called a QTI-subgroup if C _G (x) ≤ N _G (H) for any 1 ≠ x ∈ H. In this article, the finite groups all of whose second maximal subgroup are QTI-subgroups are classified.     

On the Constructive Inverse Problem in Differential Galois Theory#

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G ⁰ = G ₁ · ··· · G _r , where each G _i is a simple group of type A_?, C_?, D_?, E₆, or E₇, we construct a differential equation over C(x) having Galois group G.     

New Characterizations of p-Supersolubility of Finite Groups

Let H be a subgroup of a finite group G. H is said to be λ-supplemented in G if G has a subgroup T such that G = HT and H ∩ T ≤ H _SE , where H _SE denotes the subgroup of H generated by all those subgroups of H, which are S-quasinormally embedded in G. In this article, some results about the λ-supplemented subgroups are obtained, by which we determine the structure of some classes of finite groups. In particular, some new characterizations of p-supersolubility of finite groups are given under the assumption that some primary subgroups are λ-supplemented. As applications, a number of previous known results are generalized.     

On Π-Supplemented Subgroups of a Finite Group

A subgroup H of a finite group G is said to satisfy Π-property in G if for every chief factor L/K of G, |G/K: N_G/K(HK/K ∩ L/K)| is a π(HK/K ∩ L/K)-number. A subgroup H of G is called Π-supplemented in G if there exists a subgroup T of G such that G = HT and H ∩ T ≤ I ≤ H, where I satisfies Π-property in G. In this article, we investigate the structure of a finite group G under the assumption that some primary subgroups of G are Π-supplemented in G. The main result we proved improves a large number of earlier results.     

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.     

Enumerating Groups Acting Regularly on a d-Dimensional Cube

Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G _p  × B a finite group, where G _p is a p-group and B is a p′-group. Denote by K ^λ G a twisted group algebra of G over K with a 2-cocycle λ ∈Z ²(G, K*). In this article, we give necessary and sufficient conditions for K ^λ G to be of OTP representation type, in the sense that every indecomposable K ^λ G-module is isomorphic to the outer tensor product V#W of an indecomposable K ^λ G _p -module V and an irreducible K ^λ B-module W.