scholarly journals Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with a Natural Extension to Infinite Structures

2020 ◽  
Author(s):  
Juan Ramirez
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Linear Order ◽  
Natural Extension ◽  
Natural Numbers ◽  
Real Numbers ◽  
Tree Structures ◽  
Mathematical Objects ◽  
Finite Group Theory ◽  
Block Form

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. There is also a canonical order for the elements of $G$ and we can define equivalent objects of $G$. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. We show how to find all groups of order $n$, and order them. Examples are given using all groups with order smaller than $10$. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, as well. We make brief comments on treating the calculus of real numbers. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational and physical aspects of these representations.

scholarly journals Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with A Natural Extension To Infinite Structures

2021 ◽  
Author(s):  
Juan Pablo Ramírez
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Linear Order ◽  
Natural Extension ◽  
Natural Numbers ◽  
Real Numbers ◽  
Tree Structures ◽  
Mathematical Objects ◽  
Finite Group Theory ◽  
Block Form

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. Internally, there is also a canonical order for the elements of any finite group $G$, and we find equivalent objects. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. Examples are given, using all groups with less than ten elements, to illustrate the procedure for finding all groups of $n$ elements, and we order them externally and internally. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, also. We make brief mention on the calculus of real numbers. In general, we are able to represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects of all types are well assigned to tree structures. We conclude with comments on type theory and future work on computational and physical aspects of these representations.

scholarly journals Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with a Natural Extension to Infinite Structures

2020 ◽  
Author(s):  
Juan Ramirez
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Linear Order ◽  
Natural Extension ◽  
Natural Numbers ◽  
Real Numbers ◽  
Tree Structures ◽  
Mathematical Objects ◽  
Finite Group Theory ◽  
Block Form

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. There is also a canonical order for the elements of $G$ and we can define equivalent objects of $G$. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. We show how to find all groups of order $n$, and order them. Examples are given using all groups with order smaller than $10$. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, as well. We make brief comments on treating the calculus of real numbers. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational and physical aspects of these representations.

scholarly journals Canonical Description of Group Theory: A Linear Order on All Finite Groups

2020 ◽  
Author(s):  
Juan Pablo Ramirez
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Finite Groups ◽  
Linear Order ◽  
Natural Numbers ◽  
Real Numbers ◽  
Prime Factorization ◽  
Tree Structures ◽  
Mathematical Objects ◽  
Block Form

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order, on the quotient space of isomorphism classes of finite groups, that is well behaved with respect to cardinality. If $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G\leq\mathbb Z_{p_1}^{n_1}\oplus\mathbb Z_{p_2}^{n_2}\oplus\cdots\oplus\mathbb Z_{p_k}^{n_k}$ where $n=p_1^{n_1}p_2^{n_2}\cdots p_{k}^{n_k}$ is the prime factorization of $n$. We find a canonical order for the objects of $G$ and define equivalent objects of $G$, thus finding the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and we are provided with a minimal set of independent equations that define the group. We show how to find all groups of order $n$, and order them. We give examples using all groups with order smaller than $10$, and we find the canonical block form of the symmetry group $\Delta_4$. In the next section, we extend our results to the infinite case, which defines a real number as an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, as well. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational aspects of these representations.

scholarly journals Canonical Description of Group Theory: A Linear Order on All Finite Groups

2020 ◽  
Author(s):  
Juan Pablo Ramirez
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Natural Number ◽  
Finite Groups ◽  
Linear Order ◽  
Natural Numbers ◽  
Real Numbers ◽  
Good Definition ◽  
Mathematical Objects ◽  
Real Functions

We provide a construction of natural numbers that is unique with respect to other constructions, and use this construction in the domain of algebra and finite functions to find several results in finite group theory. First, we give a linear order to the set of all finite functions. This gives a linear order in the subset of all finite permutations. To do this, we assign a unique natural number, $N_f$, to every finite function $f$. The sub order on permutations is well defined with respect to cardinality; if $\eta_m,\eta_n$ are permutations on $m<n$ objects, then $N_{\eta_m}<N_{\eta_n}$. This representation also has the characteristic $N_{\textbf{1}_n}<N_{\eta}<N_{\textbf{id}_n}$ where $\textbf{1}_n$ is the one-cycle permutation of $n$ objects, $\textbf{id}_n$ is the identity permutation of $n$ objects, and $\eta$ is any permutation of $n$ objects. This representation provides a good definition of equivalent functions, and equivalent objects on functions. We are able to do this for both concrete functions, and abstract functions. We use this in the main section, on group theory, to number the set of all finite groups. We are able to well represent every finite group as a natural number; two groups are represented by the same natural number if and only if they are in the same isomorphism class. In fact, we are able to give a linear order to the set of finite groups. Specifically, we give a canonical bijective function $\textbf{G}_{Fin}\rightarrow\mathbb N$. This representation, $N_G$, of $G$, is also well behaved with respect to cardinality. Additionally, the cyclic group $\mathbb Z_n$ has smaller representation than any group of $n$ objects, and the group with largest representation is the abelian group $\mathbb Z_{p_1}^{n_1}\oplus\mathbb Z_{p_2}^{n_2}\oplus\cdots\oplus\mathbb Z_{p_k}^{n_k}$, where $n=p_1^{n_1}p_2^{n_2}\cdots p_{k}^{n_k}$ is the prime factorization of $n$. This representation of a finite group as a natural number also provides a linear order to the elements of the group, arranging its Cayley table in a canonical block form. The last section is an introductory description of real numbers as infinite sets of natural numbers. Real functions are represented as sets of real numbers, and sequences of real functions $f_1,f_2,\ldots$ are well represented by sets of real numbers, as well. In the last section we well assign mathematical objects to tree structures and conclude with some brief comments on type theory and future work. In general we are able to represent and manipulate mathematical objects with the smallest possible type, and minimum complexity.

scholarly journals Canonical Description of Group Theory: A Linear Order on All Finite Groups

2020 ◽  
Author(s):  
Juan Pablo Ramirez
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Natural Number ◽  
Finite Groups ◽  
Linear Order ◽  
Natural Numbers ◽  
Real Numbers ◽  
Good Definition ◽  
Mathematical Objects ◽  
Real Functions

We provide a construction of natural numbers that is unique with respect to other constructions, and use this construction in the domain of algebra and finite functions to find several results in finite group theory. First, we give a linear order to the set of all finite functions. This gives a linear order in the subset of all finite permutations. To do this, we assign a unique natural number, $N_f$, to every finite function $f$. The sub order on permutations is well defined with respect to cardinality; if $\eta_m,\eta_n$ are permutations on $m<n$ objects, then $N_{\eta_m}<N_{\eta_n}$. This representation also has the characteristic $N_{\textbf{1}_n}<N_{\eta}<N_{\textbf{id}_n}$ where $\textbf{1}_n$ is the one-cycle permutation of $n$ objects, $\textbf{id}_n$ is the identity permutation of $n$ objects, and $\eta$ is any permutation of $n$ objects. This representation provides a good definition of equivalent functions, and equivalent objects on functions. We are able to do this for both concrete functions, and abstract functions. We use this in the main section, on group theory, to number the set of all finite groups. We are able to well represent every finite group as a natural number; two groups are represented by the same natural number if and only if they are in the same isomorphism class. In fact, we are able to give a linear order to the set of finite groups. Specifically, we give a canonical bijective function $\textbf{G}_{Fin}\rightarrow\mathbb N$. This representation, $N_G$, of $G$, is also well behaved with respect to cardinality. Additionally, the cyclic group $\mathbb Z_n$ has smaller representation than any group of $n$ objects, and the group with largest representation is the abelian group $\mathbb Z_{p_1}^{n_1}\oplus\mathbb Z_{p_2}^{n_2}\oplus\cdots\oplus\mathbb Z_{p_k}^{n_k}$, where $n=p_1^{n_1}p_2^{n_2}\cdots p_{k}^{n_k}$ is the prime factorization of $n$. This representation of a finite group as a natural number also provides a linear order to the elements of the group, arranging its Cayley table in a canonical block form. The last section is an introductory description of real numbers as infinite sets of natural numbers. Real functions are represented as sets of real numbers, and sequences of real functions $f_1,f_2,\ldots$ are well represented by sets of real numbers, as well. In the last section we well assign mathematical objects to tree structures and conclude with some brief comments on type theory and future work. In general we are able to represent and manipulate mathematical objects with the smallest possible type, and minimum complexity.

The Problem of Mathematical Objects

2020 ◽  
pp. 213-224
Author(s):  
Bob Hale
Keyword(s):  
Natural Numbers ◽  
Real Numbers ◽  
Mathematical Objects ◽  
Epistemological Foundation ◽  
Fundamental Part ◽  
Object Based

If fundamental mathematical theories such as arithmetic and analysis are taken at face value, any attempt to provide an epistemological foundation—roughly, an account which explains how we can know standard mathematical theories to be true, or at least justifiably believe them—must confront the problem of mathematical objects—the problem of explaining how a belief in the existence of an infinity of natural numbers, an uncountable infinity of real numbers, etc., is to be justified. One small but fundamental part of the problem is discussed: whether we can be justified in believing that there is a denumerable infinity of natural numbers, or, more generally, an infinity of objects of any kind. The chapter considers two broad approaches to this problem—what are called object-based and property-based approaches.

scholarly journals SIEVES AND THE MINIMAL RAMIFICATION PROBLEM

2018 ◽  
Vol 19 (3) ◽  
pp. 919-945 ◽  
Author(s):  
Lior Bary-Soroker ◽  
Tomer M. Schlank
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Upper Bounds ◽  
Arithmetic Geometry ◽  
Linear Forms ◽  
Quantitative Version ◽  
Finite Group Theory ◽  
Inverse Galois Problem ◽  
Branched Covering ◽  
Image Position

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a $G$-Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$.In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$ for all $n,k>0$. Here $S_{k}$ denotes the symmetric group on $k$ letters, and $S_{k}^{n}$ is the direct product of $n$ copies of $S_{k}$. We also get the correct asymptotic of $m(G^{n})$, as $n\rightarrow \infty$ for a certain class of groups $G$.Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

Constructive assertions in an extension of classical mathematics

1982 ◽  
Vol 47 (2) ◽  
pp. 359-387 ◽  
Author(s):  
Vladimir Lifschitz
Keyword(s):  
Subject Matter ◽  
Formal Languages ◽  
Natural Numbers ◽  
Real Numbers ◽  
Mathematical Objects ◽  
Classical Mathematics ◽  
The Mind ◽  
The Universe

We distinguish between two kinds of mathematical assertions: objective and constructive. An objective assertion describes the universe of mathematical objects; a constructive one describes the (idealized) mathematician's ability to find mathematical objects with various properties. The familiar formalizations of classical mathematics are based on formal languages designed for expressing objective assertions only. The constructivist program stresses, on the contrary, the importance of constructive assertions; moreover, intuitionism claims that constructive activities of the mind constitute the very subject matter of mathematics, and thus questions the semantic status of objective assertions.The purpose of this paper is to show that classical mathematics can be extended to include constructive sentences, so that both objective and constructive properties can be discussed in the framework of the same theory. To achieve this goal, we introduce a new property of mathematical objects, calculability.The word “calculable” may be applied to objects of various types: natural numbers, integers, rational or real numbers, polynomials with rational or real coefficients, etc. In each case it has a different meaning, so that actually we define not one, but many new properties.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

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