automorphism

简明释义

英[ˌɔːtəʊˈmɔːfɪzəm]美[ˌɔtəˈmɔrfɪzəm]

n. [数] 自同构；自守

英英释义

单词用法

同义词

反义词

例句

1.In Chapter 2, we investigate the longest elements and the corresponding special automorphism of finite reflection groups.

第二章，主要研究有限反射群的最长元以及和最长元相对应的有限反射群的自同构。

2.By using the theories in basic algebra about automorphism, left translation and normal subgroup, in the holomorph of G is discussed briefly, and several related conclusions are obtained.

运用基础代数中有关自同构、左平移、正规子群等理论，对群G的全形进行了简单的探讨，证明了几个有关的结论。

3.In other words it has a large automorphism group.

换言之，它有大的自同构群。

4.We can distinguish nonlinear automorphism of polynomials algebra by using the test polynomials.

给出了一类新的试验多项式，可识别多项式代数的非线性自同构。

5.It is often very difficult to identify the structure of the automorphism group and the inner automorphism group of a group, and there is no general theory and method.

确定一个群的自同构群和内自同构群的结构往往十分困难，还没有一般性的理论及方法。

6.We introduce the conception of involutorial anti automorphism over distributive pseudolattices, define and get some properties of M-P inverse of matrix.

在分配伪格上引入对合反自同构和矩阵M-P逆的概念，得到矩阵M-P逆的若干性质。

7.The set of all automorphisms of a mathematical structure forms a group under composition.

一个数学结构的所有自同构的集合在复合下形成一个群。

8.In group theory, an automorphism is a mapping from a group to itself that preserves the group structure.

在群论中，自同构是从一个群到其自身的映射，它保持群的结构。

9.The study of automorphisms can help us understand the symmetries of mathematical objects.

研究自同构可以帮助我们理解数学对象的对称性。

10.In topology, a homeomorphism is a special type of automorphism that preserves topological properties.

在拓扑学中，同胚是一种特殊类型的自同构，保持拓扑性质不变。

11.An automorphism of a graph is a permutation of its vertices that preserves the adjacency relation.

图的自同构是其顶点的一个排列，保持邻接关系不变。

作文

In the field of mathematics, particularly in abstract algebra and geometry, the concept of automorphism is fundamental. An automorphism can be defined as a special type of isomorphism that maps an object to itself while preserving its structure. This means that if we have a mathematical object, such as a group, a ring, or a geometric figure, an automorphism will maintain the relationships and properties of that object during the mapping process. Understanding automorphism is crucial for mathematicians because it helps them analyze the symmetries and inherent structures within various mathematical systems.To illustrate the concept of automorphism, let’s consider a simple example from group theory. Suppose we have a group G, which consists of elements that combine according to specific rules. An automorphism of this group is a bijective function from G to itself that respects the group operation. This means that if you take any two elements a and b from G, the automorphism will satisfy the condition: f(a * b) = f(a) * f(b), where * denotes the group operation and f is the automorphism. This property ensures that the structure of the group is preserved under the function f.Moreover, automorphisms can provide valuable insights into the nature of the mathematical object they are associated with. For instance, the set of all automorphisms of a given group forms a new group known as the automorphism group. This new group captures the essence of how the original group can be transformed while retaining its internal structure. By studying this automorphism group, mathematicians can gain a deeper understanding of the symmetries and characteristics of the original group.In the realm of geometry, automorphisms can also be seen in the study of geometric figures. For example, consider a square. The automorphisms of a square include rotations and reflections that map the square onto itself. Each of these transformations preserves the distances and angles of the square, demonstrating how automorphisms operate in a geometric context. These transformations are essential in understanding the symmetry properties of shapes and can lead to broader applications in fields such as physics and engineering.Furthermore, the concept of automorphism extends beyond pure mathematics and finds applications in various scientific disciplines. In computer science, for example, automorphisms are relevant in graph theory, where they help in analyzing the symmetries of graphs. Understanding the automorphisms of a graph can assist in optimizing algorithms and improving computational efficiency.In conclusion, the notion of automorphism is a powerful tool in mathematics that allows us to explore the symmetries and structures of various objects. Whether in abstract algebra, geometry, or even computer science, automorphisms play a crucial role in understanding how different elements relate to one another while preserving their inherent properties. As we delve deeper into the world of mathematics, the significance of automorphism becomes increasingly clear, showcasing the beauty and complexity of mathematical relationships.

在数学领域，尤其是在抽象代数和几何中，自同构的概念是基础性的。自同构可以被定义为一种特殊类型的同构，它将一个对象映射到自身，同时保持其结构。这意味着，如果我们有一个数学对象，例如一个群、一个环或一个几何图形，自同构将在映射过程中保持该对象的关系和属性。理解自同构对数学家来说至关重要，因为它帮助他们分析各种数学系统中的对称性和内在结构。为了说明自同构的概念，让我们考虑一个来自群论的简单例子。假设我们有一个群G，它由按照特定规则组合的元素组成。这个群的一个自同构是一个从G到自身的双射函数，尊重群运算。这意味着，如果你取群G中的任意两个元素a和b，自同构将满足条件：f(a * b) = f(a) * f(b)，其中*表示群运算，f是自同构。这个性质确保了在函数f下群的结构得以保留。此外，自同构可以提供关于与之相关的数学对象的宝贵见解。例如，给定群的所有自同构的集合形成一个新的群，称为自同构群。这个新群捕捉了原始群在保留其内部结构的情况下如何被转换。通过研究这个自同构群，数学家可以更深入地理解原始群的对称性和特征。在几何学领域，自同构也可以在几何图形的研究中看到。例如，考虑一个正方形。正方形的自同构包括将正方形映射到自身的旋转和反射。每一个这些变换都保持了正方形的距离和角度，展示了自同构在几何上下文中的运作方式。这些变换对于理解形状的对称性特性至关重要，并且可以导致物理和工程等领域的更广泛应用。此外，自同构的概念超越了纯数学，并在各个科学学科中找到了应用。在计算机科学中，例如，自同构在图论中是相关的，它们帮助分析图的对称性。理解图的自同构可以辅助优化算法并提高计算效率。总之，自同构的概念是数学中的一个强大工具，使我们能够探索各种对象的对称性和结构。无论是在抽象代数、几何还是计算机科学中，自同构在理解不同元素之间的关系时起着至关重要的作用，同时保持其固有属性。随着我们深入数学的世界，自同构的重要性变得愈加明显，展示了数学关系的美丽和复杂性。