Elementary equivalence of endomorphism rings of Abelian p-groups

In this paper we study a relationship between elementary equivalence of endomorphism rings of Abelian p-groups and second-order equivalence of the corresponding Abelian p-groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 135–224, 2004.     

On a Note From Oliver Concerning Generalized Euclidean Group Rings

In an unpublished 1987 letter, Bob Oliver determined which elementary abelian 2-groups have generalized euclidean integral group rings. He produced a filtration of E ₂(R) by normal subgroups, sandwiched between elementary and special linear relative groups, with layers that are second homology groups with mod-2 coefficients. His proof is presented here, with related consequences for some other finite groups.     

Torsion in Mordell-Weil groups of Fermat Jacobians

We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent p over the cyclotomic field obtained by adjoining a primitive p-th root of 1 to Q. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian p-groups.     

Connected components of the category of elementary abelian p-subgroups

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p+1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1,2,p+1, or possibly 3 classes for some 3-groups of maximal nilpotency class.     

On the action of the Iwahori–Hecke algebra on modular invariants

We compute the action of the modular Iwahori–Hecke algebra on the ring of invariants of the mod p cohomology of elementary p-groups under Borel subgroup of the general linear group. Applications include a direct proof of the structure of the universal Steenrod algebra and a new proof of a key result on the structure of the Takayasu modules.     

Finite quaternion-free 2-groups

We present an elementary proof of the classification theorem for finite nonmodular quaternion-free 2-groups. This proof does not involve the structure theory of powerful 2-groups. Such a new proof is also necessary, since there are several gaps in the original proof given in [5].     

Subgroups of powerful groups

We consider subgroups of powerfulp-groups. In particular, we give a new proof that allp-groups are sections of powerfulp-groups, give necessary and sufficient conditions for a 2-generator group to be a normal subgroup of a powerfulp-groups, and show thatp-groups of class 2, orp-groups with a cyclic commutator subgroup, are such normal subgroups.     

Torsionsgruppen,deren Untergruppen alle subnormal sind

We prove that a group, which is the extension of a nilpotent torsion group by a soluble group of finite exponent and all of whose subgroups are subnormal, is nilpotent. The problem can be easily reduced to the investigation of extensions of abelian torsion groups by elementary abelian p-groups with all subgroups of these extensions subnormal.     

Varieties and Torsion Classes of m-Groups

We prove that every variety of m-groups is a torsion class; find basis of identities for a product variety of m-groups; and show that the product of every finitely based variety of m-groups and a variety of Abelian m-groups is a finitely based variety.     

A Classification of Regular p-Groups with Invariants (e, 2, 1)

The class of the regular p-groups is one of the important classes in p-groups. Not only it has many similar properties as abelian p-groups, but also many of the p-groups belong to this class. In this paper, using the algorithms for determining the isomorphic regular p-groups, we give a complete classification of the regular p-groups with e-invariants (e, 2, 1).Supported by SXYSF 991003.     

Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.      

Normalizers of maximal tori

Normalizers and p-normalizers of maximal tori in p-compact groups can be characterized by the Euler characteristic of the associated homogeneous spaces. Applied to centralizers of elementary abelian p-groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus; i.e. that “normalizer” commutes with “centralizer”. Received April 1, 1995; in final form August 11, 1997     

Elementary abelian p-subgroups of algebraic groups

Let be an algebraically closed field and let G be a finite-dimensional algebraic group over which is nearly simple, i.e. the connected component of the identity G ⁰ is perfect, C _G(G ⁰)=Z(G ⁰) and G ⁰/Z(G ⁰) is simple. We classify maximal elementary abelian p-subgroups of G which consist of semisimple elements, i.e. for all primes p char .Call a group quasisimple if it is perfect and is simple modulo the center. Call a subset of an algebraic group toral if it is in a torus; otherwise nontoral. For several quasisimple algebraic groups and p=2, we define complexity, and give local criteria for whether an elementary abelian 2-subgroup of G is toral.For all primes, we analyze the nontoral examples, include a classification of all the maximal elementary abelian p-groups, many of the nonmaximal ones, discuss their normalizers and fusion (i.e. how conjugacy classes of the ambient algebraic group meet the subgroup). For some cases, we give a very detailed discussion, e.g. p=3 and G of type E ₆, E ₇ and E ₈. We explain how the presence of spin up and spin down elements influences the structure of projectively elementary abelian 2-groups in Spin(2n, ). Examples of an elementary abelian group which is nontoral in one algebraic group but toral in a larger one are noted.Two subsets of a maximal torus are conjugate in G iff they are conjugate in the normalizer of the torus; this observation, with our discussion of the nontoral cases, gives a detailed guide to the possibilities for the embedding of an elementary abelian p-group in G. To give an application of our methods, we study extraspecial p-groups in E ₈( ).Dedicated to Jacques Tits for his sixtieth birthday     

Wreath products of the groups of monotone permutations

We introduce the concept of wreath product of the m-groups of permutations and prove that an m-transitive group of permutations with an m-congruence is embeddable into the wreath product of the suitable m-transitive m-groups of permutations. This implies that an arbitrary m-transitive group in the product of two varieties of m-groups embeds into the wreath product of the suitable m-transitive groups of these varieties.     

Quasi-Boolean Powers of Elementary 2-Groups

Groupoids with unit that admit embeddings into quasi-Boolean powers of elementary 2-groups are characterized.     

Finite Soluble Groups with Permutable Subnormal Subgroups

A finite group G is said to be a PST-group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST-groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT-groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PST-groups. Our techniques and results provide a unified point of view for T-groups, PT-groups, and PST-groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure.     

On Components of the Partial Schur Multiplier

In this paper we study properties of the partial Schur multiplier. As an application a characterization of components of the partial Schur multiplier for some groups is given. We also describe the classical Schur multiplier of elementary abelian 2-groups by using partial factor sets.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008