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Register a new userRings associated to coverings of finite p-groups
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In general the endomorphisms of a non-abelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group which are endomorphisms when restricted to the elements of a cover of the group by abelian subgroups. We give an algorithm which allows us to determine the elements of the ring of functions of a finite $p$-group which arises in this manner when the elements of the cover are required to be either cyclic or elementary abelian of rank $2$. This enables us to determine the actual structure of such a ring as a subdirect product. A key part of the argument is the construction of a graph whose vertices are the subgroups of order $p$ and whose edges are determined by the covering.
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We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring $Z G$ of a finite nilpotent group $G$ has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups $(H,K)$, called strong Shoda pairs, and explicit constructed central elements $e(G,H,K)$ play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of such $e(G,H,K)$, with $(H,K)$ strong Shoda pairs in subgroups of $G$.
The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $mathbb Z G$ , i.e. the prime graph of the normalised unit group of $mathbb Z G$ coincides with that one of the group $G$. In this note we prove for finite groups $G$ a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups $G$ whose order is divisible by at most three primes and show that the Gruenberg - Kegel graph of such groups coincides with the prime graph of $G$.
In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra $A$, we show that the finite generation condition on $A$-modules can be replaced by a condition on any affine commutative $A$-module algebra $R$ under the assumption that $R$ is integral over its invariant subring $R^A$. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if $A$ is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra $A$ over a finite field, where $A$ can be viewed as a deformation of $A$, and prove that if the finite generation condition holds for $A$, then the same condition holds for $A$.
For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this paper, working over an algebraically closed field $mathbf{k}$ of prime characteristic $p$, we introduce the concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional connected Hopf algebras which are almost primitively generated; that is, these connected Hopf algebras are $p^{n+1}$-dimensional, whose primitive spaces are abelian restricted Lie algebras of dimension $n$. We illustrate this technique for the case $n=2$. Together with our preceding results in arXiv:1309.0286, we provide a complete classification of $p^3$-dimensional connected Hopf algebras over $mathbf{k}$ of characteristic $p>2$.
Using the Luthar--Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups $Co_1$, $Co_2$ and $Co_3$.
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