Isomorphism of Modular Group Algebras of p-Mixed Abelian Groups

Let A be a torsion-free abelian group and F a free subgroup of A. We prove that if A/F is a reduced p-group and A/(F + C) is reduced for every p-pure subgroup C of A, then A is free.

Let KG be the group algebra of an abelian group G over a field K of prime characteristic p. Denote by S(KG) the p-component of the group V(KG) of normalized units of KG (of augmentation 1). Let H be an arbitrary group and KH ? KG as K-algebras. We prove the following. First, assume that G is a splitting group, the p-component G _p of G is simply presented, and the field K is perfect. Then H _p  ? G _p . If, in addition, G is p-mixed, then G _p is a direct factor of S(KG), and G is a direct factor of V(KG), each with the same simply presented complement. Secondly, we introduce a class of special p-mixed abelian groups and prove that, if G belong to this class, then any group basis of the group algebra KG splits. Besides, H is p-mixed and splits. Thirdly, if G is a special p-mixed abelian group and G _p is a reduced totally projective p-group, then H ? G. These results correct some essential inaccuracies and incompleteness in the proofs of results in this direction of Danchev [3-8 Danchev , P. V. ( 1998 ). Isomorphism of commutative group algebras of mixed splitting groups . Compt. Rend. Acad. Bulg. Sci. 51 : 13 – 16 . Danchev , P. V. ( 2000 ). Isomorphism of modular group algebras of totally projective abelian groups . Communications in Algebra 28 : 2521 – 2531 . Danchev , P. V. ( 2001 ). On a question of W. L. May concerning the isomorphism of modular group algebras . Communications in Algebra 29 : 1953 – 1958 . Danchev , P. V. ( 2001 ). Normed units in Abelian group rings . Glasg. Math. J. 43 : 365 – 373 . Danchev , P. V. ( 2002 ). Invariants for group algebras of splitting abelian groups with simply presented components . Compt. Rend. Acad. Bulg. Sci. 55 : 5 – 8 . Danchev , P. V. ( 2004 ). A note on the isomorphic modular group algebras of abelian groups with simply presented p-components . Compt. Rend. Acad. Bulg. Sci. 57 : 13 – 14 . ].     

Commutative Group Algebras of Cardinality {\cal N} 1

We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define G_p and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if G_p is Hausdorff and both F and G have cardinalities not exceeding ₁, then S(FG)/G_p is a direct sum of cyclics. Thus G_p is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power ₁ whose p-component G_p is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between G_p and H_p. In particular, when G is of power ₁ and is p-mixed of torsion-free rank 1 whose G_p is torsion-complete, we have G H. If F is in power p and G, with cardinality ₁, is a direct sum of p-local algebraically compact groups such that FG FH as F-algebras for some group H, then G H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.     

Estimating Additive Character Sums for Fuchsian Groups

In the usual construction of non-holomorphic Eisenstein series, for a general Fuchsian group, a multiplicative character may be included. The properties of these series are well known. Here we instead include an additive character and develop the properties of the resulting series. We pay particular attention to additive characters that are non-cuspidal, i.e., that are not zero on some parabolic generators. These series may be used to estimate certain additive character sums. For example, asymptotics for a weighted sum over group elements that counts the number of appearances of a fixed generator of the Fuchsian group are obtained.     

The Subgroups of Direct Products of Surface Groups

A subgroup of a product of n surface groups is of type FP _n if and only if it contains a subgroup of finite index that is itself a product of (at most n) surface groups.     

Notes on Essentially Finitely Indecomposable Nonthick Primary Abelian Groups

Let R be a commutative Noetherian ring of prime characteristic and M be an x-divisible right R[x, f]-module that is Noetherian as R-module. We give an affirmative answer to the question of Sharp and Yoshino in the case where R is semilocal and prove that the set of graded annihilators of R[x, f]-homomorphic images of M is finite. We also give a counterexample in the general case.     

Simplicity of Skew Group Rings of Abelian Groups

Necessary and sufficient conditions for simplicity of a general skew group ring A ?_σ G are not known. In this article, we show that a skew group ring A ?_σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ?_σ G is simple.     

The Derivation Problem for Group Algebras of Connected Locally Compact Groups

The derivation problem for a locally compact group G is to decidewhether for each derivation D from L¹(G) into L¹(G) there isa bounded measure µM(G) with D(a) = aµ–µa(a L¹(G)). In this paper we obtain an affirmative answer forthe case of connected groups. To explain the contents of thispaper we give an equivalent formulation of the problem. Supposethat the group G acts as a group of homeomorphisms of the locallycompact space X. Related to this there is an action of G onM(X). A bounded crossed homomorphism from G to M(X) is a map with bounded range and satisfying (gh) = g(h)+(g) (g, h G).The problem for bounded crossed homomorphisms is to decide iffor each such there is an element µ of M(X) with (g)= gµ– µ (g G). The derivation problem isequivalent to this bounded crossed homomorphism problem forthe special case X = G where G acts on X by conjugation (togetherwith some mild continuity hypotheses about the map :GM(X) whichare often automatically satisfied). The bounded crossed homomorphismproblem always has a positive solution if G is amenable anda closely related calculation shows that in solving the boundedcrossed homomorphism problem we need only solve it for functions which are zero on H where H is a given amenable subgroup ofG. It can happen that this condition of being zero on H forces to be zero even when H is a comparatively small subgroup ofG. If h is an element of G such that ‘hⁿx ’ asn for all x X then for any two measures µ and , forlarge values of n, µ and hⁿ have little overlap so ||µ+ hⁿ|| ||µ|| + ||||. Thus if H is the subgroup generatedby h, for any g G .     

Commutative Group Algebras of Summable p-Groups

We study numerical semigroups generated by generalized arithmetic sequences. We present a membership criterion for such a numerical semigroup, and by this we are able to answer fundamental questions concerning a numerical semigroup such as computing the Frobenius number and the type of the numerical semigroup, and decide whether the numercial semigroup is symmetric. Also for this kind of numerical semigroups, we compute the cardinality of a minimal presentation and determine whether they are complete intersections.     

Generalized Group Algebras of Locally Compact Groups

This article studies the homological properties of generalized group algebra L ¹(G, A) of a locally compact group G over a Banach algebra A with an identity of norm 1. It is shown that if L ¹(G, A) is right continuous then G is finite and A is right continuous. It is also shown that L ¹(G, A) is right self-injective if and only if G is finite and A is right self-injective.     

Isomorphic Commutative Group Algebras of p-Mixed Warfield Groups

Let G be a p-mixed Warfield Abelian group and F a field of char F = p ≠ 0. It is proved that if for any group H the group algebras FH and FG are F-isomorphic, then H is isomorphic to G. This presentation enlarges a result of W. May argued when G is p-local Warfield Abelian and published in Proc. Amer. Math. Soc. （1988）.     

Maximal Divisible Subgroups in p-Mixed Modular Abelian Group Rings

The isomorphism structure of the maximal divisible subgroup dVR(G) of the normed unit group VR(G) in a commutative group ring R(G) is completely described only in terms of R and G whenever R is a commutative unital ring of prime characteristic p and G is a p-mixed abelian group. This somewhat extends a result due to Nachev [13 Nachev , N. A. ( 1995 ). Invariants of the Sylow p-subgroup of the unit group of a commutative group ring of characteristic p. Commun. Algebra 23 : 2469 – 2489 . [Google Scholar]] as well as results due to the author [1 Danchev , P. V. ( 2004 ). Maximal divisible subgroups in modular group algebras of p-mixed and p-splitting abelian groups . Rad. Mat. 13 : 23 – 32 . [Google Scholar]] and Kuneva [11 Kuneva , V. ( 2006 ). Maximal divisible subgroups in modular group algebras of p-mixed abelian groups . Compt. rend. Acad. bulg. Sci. 59 : 899 – 902 . [Google Scholar]].     

On the Unit Groups of Commutative Modular Group Algebras of KT-Groups

ABSTRACT

In this article, we prove that the inner projection of a projective curve with higher linear syzygies has also higher linear syzygies. Specifically, if a very ample line bundle ? on a smooth projective curve X satisfies property N _p for p  ≥  1 and H ¹ (? ^? 2) =  0 , then ?( ?  q ) satisfies property N _{p ? 1} for any point q  ∈  X . We also give simple proofs of well-known theorems about syzygies and raise some questions related to the line bundles of degree 2 g which do not satisfy property N ₁ .     

Local Subgroups and Group Algebras of Finite p-Solvable Groups

Let p be a prime, k an algebraically closed field ofcharacteristic p, and G a finite group with a Sylowp-Subgroup P. In this paper, we consider the property thatNG(P)/P is Abelian. We provide somenecessary or sufficient conditions NG(P)/P to be Abelian in term of thestructure of the group algebra kG as a k-algebra, in casethat G is p-nilpotent or of p-length 1.     

The k-Orbit Reconstruction for Abelian and Hamiltonian Groups

Let G be a permutation group on a set . Then G acts in the natural way on the collection ^{k} of all k-element subsets. Orbits under this action are called k-orbits. A problem similar to the Edge-Reconstruction Conjecture in graph theory can be posed for k-orbits of a general group G. Here the k-orbit reconstruction problem is solved for transitive Abelian and Hamiltonian groups: all k-orbits of Abelian groups are reconstructible if k>3 and the same is true for Hamiltonian groups if k>4.     

Lifting Property of Direct Sums of Hollow Modules

We give necessary and sufficient conditions for a module M with a decomposition that complements maximal direct summands to be a lifting module. We apply our results to get a necessary and sufficient condition for a direct sum of modules to be a quasi-discrete (discrete) module. We also give some sufficient conditions for a lifting module to be a direct sum of indecomposables.     

Inductive Algebras for Finite Heisenberg Groups

A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite Heisenberg groups.     

The Rank of a Direct Power of a Small-Cancellation Group

We construct an example of a finitely generated group G such that rank((G )ⁿ)=2 for all n1. For each n, we construct a finitely presented group G _n such that rank((G _n )ⁿ)=2. We conjecture that if G is a word-hyperbolic group then rank(G ⁿ ) as $ n. For each m we give an example of a residually finite group K _m such that K _m has exactly two relators, but K _m has no proper subgroups of index $ m. We construct a finitely generated group D such that there is an epimorphism DD×D.     

关于有限Abel群直接因子的平均数

对于有限Ａｂｅｌ群Ｇ，ｔ（Ｇ）定义为其直接因子的个数，在［５］中，Ｅ．Ｋｒａｔｚｅｌ对于和式得到一渐近公式，对应余项为△＿１（ｘ）．本文利用Ｅ．Ｆｏｕｖｒｙ和Ｈ　．Ｉｗａｎｉｅｃ的方法，得到了关于△＿１（ｘ）的一个新的估计，改进了Ｈ．Ｍｅｎｚｅｒ等人的结果．