Cyclic groups and elementary number theory
ثبت نشده
چکیده
The story of cyclic groups is very much connected with elementary number theory (factorization, prime numbers, congruences). We prove some basic facts from “first principles,” using a few results about addition, multiplication, and order in N. One of the most basic facts is the following: Theorem 1.1 (Long division with remainder). Let n ∈ N. Then for all a ∈ Z, there exist unique integers q and r with 0 ≤ r ≤ n − 1 such that a = nq + r.
منابع مشابه
A Simple Classification of Finite Groups of Order p2q2
Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, respectively. In this paper, we show that up to isomorphism, there are four groups of order p^2q^2 when Q and P are cyclic, three groups when Q is a cyclic and P is an elementary ablian group, p^2+3p/2+7 groups when Q is an elementary ablian group an...
متن کاملBeyond first order logic: From number of structures to structure of numbers: Part II
We study the history and recent developments in nonelementarymodel theory focusing on the framework of abstractelementary classes. We discuss the role of syntax and semanticsand the motivation to generalize first order model theory to nonelementaryframeworks and illuminate the study with concrete examplesof classes of models. This second part continues to study the question of catecoricitytrans...
متن کاملOn the Number of Fuzzy Subgroups of Finite Symmetric Groups
One of the most important problems of fuzzy group theory is to classify the fuzzy subgroups of a finite group. This topic has enjoyed a rapid development in the last few years. Several papers have treated the particular case of finite abelian groups. Thus, in [13] the number of distinct fuzzy subgroups of a finite cyclic group of square-free order is determined, while [14–16] and [28] deal with...
متن کاملThe Effect of Choice Theory Training to Parents on the Aggression of Their Children in Elementary School
Background and Purpose: One of the most common problems of elementary school children is aggression which has a direct relationship with parental attitudes and parenting styles. The purpose of this study was to investigate the effect of choice theory training to mothers on aggression of their children in elementary school. Method: This semi-experimental study was of pretest-posttest control gro...
متن کاملSmarandache algebras and their subgroups
In this paper we define S algebras and show that every finite group can be found in some S algebra. We define and study the S degree of a finite group and determine the S degree of several classes of finite groups such as cyclic groups, elementary abelian $p$-groups, and dihedral groups $D_p$.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2016