Some Properties of p-Groups and Commutative p-Groups

This article describes some properties of p-groups and some properties of commutative p-groups.Liang Xiquan - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Marco Riccardi. The Sylow theorems. Formalized Mathematics, 15(3):159-165, 2007, doi:10.2478/v10037-007-0018-3.Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Commutator and center of a group. Formalized Mathematics, 2(4):461-466, 1991.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Witt vectors as a polynomial functor

For every commutative ring $A$, one has a functorial commutative ring $W(A)$ of $p$-typical Witt vectors of $A$, an iterated extension of $A$ by itself. If $A$ is not commutative, it has been known since the pioneering work of L. Hesselholt that $W(A)$ is only an abelian group, not a ring, and it is an iterated extension of the Hochschild homology group $HH_0(A)$ by itself. It is natural to expect that this construction generalizes to higher degrees and arbitrary coefficients, so that one can define "Hochschild-Witt homology" $WHH_*(A,M)$ for any bimodule $M$ over an associative algebra $A$ over a field $k$. Moreover, if one want the resulting theory to be a trace theory in the sense of arXiv:1308.3743, then it suffices to define it for $A=k$. This is what we do in this paper, for a perfect field $k$ of positive characteristic $p$. Namely, we construct a sequence of polynomial functors $W_m$, $m \geq 1$ from $k$-vector spaces to abelian groups, related by restriction maps, we prove their basic properties such as the existence of Frobenius and Verschiebung maps, and we show that $W_m$ are trace functors in the sense of arXiv:1308.3743. The construction is very simple, and it only depends on elementary properties of finite cyclic groups.Comment: LaTeX2e, 49 pages. Final version -- corrected some typo

Continued fractions, modular symbols, and non-commutative geometry

Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient $PGL(2,\bold{Z})\setminus\bold{P}^1(\bold{R})$ should be considered as non-commutative modular curve, and show that the modular complex can be seen as a sequence of $K_0$-groups of the related crossed-product $C^*$-algebras. This paper is an expanded version of the previous "On the distribution of continued fractions and modular symbols". The main new features are Section 4 on non-commutative geometry and the modular complex and Section 1.2.2 on the Mixmaster Universe.Comment: AMS-TeX, 50 pages, 2 figures (eps

The radicals of semigroup algebras with chain conditions.

by Au Yun-Nam.Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 133-137).Introduction --- p.ivChapter 1 --- Preliminaries --- p.1Chapter 1.1 --- Some Semigroup Properties --- p.1Chapter 1.2 --- General Properties of Semigroup Algebras --- p.5Chapter 1.3 --- Group Algebras --- p.7Chapter 1.3.1 --- Some Basic Properties of Groups --- p.7Chapter 1.3.2 --- General Properties of Group Algebras --- p.8Chapter 1.3.3 --- Δ-Method for Group Algebras --- p.10Chapter 1.4 --- Graded Algebras --- p.12Chapter 1.5 --- Crossed Products and Smash Products --- p.14Chapter 2 --- Radicals of Graded Rings --- p.17Chapter 2.1 --- Jacobson Radical of Crossed Products --- p.17Chapter 2.2 --- Graded Radicals and Reflected Radicals --- p.18Chapter 2.3 --- Radicals of Group-graded Rings --- p.24Chapter 2.4 --- Algebras Graded by Semilattices --- p.26Chapter 2.5 --- Algebras Graded by Bands --- p.27Chapter 2.5.1 --- Hereditary Radicals of Band-graded Rings --- p.27Chapter 2.5.2 --- Special Band-graded Rings --- p.30Chapter 3 --- Radicals of Semigroup Algebras --- p.34Chapter 3.1 --- Radicals of Polynomial Rings --- p.34Chapter 3.2 --- Radicals of Commutative Semigroup Algebras --- p.36Chapter 3.2.1 --- Commutative Cancellative Semigroups --- p.37Chapter 3.2.2 --- General Commutative Semigroups --- p.39Chapter 3.2.3 --- The Nilness and Semiprimitivity of Commutative Semigroup Algebras --- p.45Chapter 3.3 --- Radicals of Cancellative Semigroup Algebras --- p.48Chapter 3.3.1 --- Group of Fractions of Cancellative Semigroups --- p.48Chapter 3.3.2 --- Jacobson Radical of Cancellative Semigroup Algebras --- p.54Chapter 3.3.3 --- Subsemigroups of Polycyclic-by-Finite Groups --- p.57Chapter 3.3.4 --- Nilpotent Semigroups --- p.59Chapter 3.4 --- Radicals of Algebras of Matrix type --- p.62Chapter 3.4.1 --- Properties of Rees Algebras --- p.62Chapter 3.4.2 --- Algebras Graded by Elementary Rees Matrix Semigroups --- p.65Chapter 3.5 --- Radicals of Inverse Semigroup Algebras --- p.68Chapter 3.5.1 --- Properties of Inverse Semigroup Algebras --- p.69Chapter 3.5.2 --- Radical of Algebras of Clifford Semigroups --- p.72Chapter 3.5.3 --- Semiprimitivity Problems of Inverse Semigroup Algebras --- p.73Chapter 3.6 --- Other Semigroup Algebras --- p.76Chapter 3.6.1 --- Completely Regular Semigroup Algebras --- p.76Chapter 3.6.2 --- Separative Semigroup Algebras --- p.77Chapter 3.7 --- Radicals of Pi-semigroup Algebras --- p.80Chapter 3.7.1 --- PI-Algebras --- p.80Chapter 3.7.2 --- Permutational Property and Algebras of Permutative Semigroups --- p.80Chapter 3.7.3 --- Radicals of PI-algebras --- p.82Chapter 4 --- Finiteness Conditions on Semigroup Algebras --- p.85Chapter 4.1 --- Introduction --- p.85Chapter 4.1.1 --- Preliminaries --- p.85Chapter 4.1.2 --- Semilattice Graded Rings --- p.86Chapter 4.1.3 --- Group Graded Rings --- p.88Chapter 4.1.4 --- Groupoid Graded Rings --- p.89Chapter 4.1.5 --- Semigroup Graded PI-Algebras --- p.91Chapter 4.1.6 --- Application to Semigroup Algebras --- p.92Chapter 4.2 --- Semiprime and Goldie Rings --- p.92Chapter 4.3 --- Noetherian Semigroup Algebras --- p.99Chapter 4.4 --- Descending Chain Conditions --- p.107Chapter 4.4.1 --- Artinian Semigroup Graded Rings --- p.107Chapter 4.4.2 --- Semilocal Semigroup Algebras --- p.109Chapter 5 --- Dimensions and Second Layer Condition on Semigroup Algebras --- p.119Chapter 5.1 --- Dimensions --- p.119Chapter 5.1.1 --- Gelfand-Kirillov Dimension --- p.119Chapter 5.1.2 --- Classical Krull and Krull Dimensions --- p.121Chapter 5.2 --- The Growth and the Rank of Semigroups --- p.123Chapter 5.3 --- Dimensions on Semigroup Algebras --- p.124Chapter 5.4 --- Second Layer Condition --- p.128Notations and Abbreviations --- p.132Bibliography --- p.13

Stabilizing bisets

Let G be a finite group and let R be a commutative ring. We analyse the (G,G)-bisets which stabilize an indecomposable RG-module. We prove that the minimal ones are unique up to equivalence. This result has the same flavor as the uniqueness of vertices and sources up to conjugation and resembles also the theory of cuspidal characters in the context of Harish-Chandra induction for reductive groups, but it is different and very general. We show that stabilizing bisets have rather strong properties and we explore two situations where they occur. Moreover, we prove some specific results for simple modules and also for p-groups