coset

简明释义

英[ˈkəʊset]美[ˈkoʊset]

n. [数] 陪集

英英释义

单词用法

同义词

反义词

例句

1.The coset model belongs to the model of the noncritical type MT, witch is never carefully studied before in documents.

该陪集模型属于非临界的MT型模型，尚未有文献仔细研究过。

2.Coset code of write run-length limited memory is analysed here.

同时对游程限制书写记忆介质的陪集码构造方法进行了分析。

3.They may start and end at the same coset.

它们可以开始和终止于同一陪集。

4.The coset model belongs to the model of the noncritical type MT, witch is never carefully studied before in documents.

该陪集模型属于非临界的MT型模型，尚未有文献仔细研究过。

5.To find the number of distinct cosets 陪集, you can use Lagrange's theorem.

要找到不同的陪集的数量，你可以使用拉格朗日定理。

6.In group theory, a coset 陪集 is formed when you multiply all elements of a subgroup by a fixed element from the group.

在群论中，当你用群中的一个固定元素乘以子群的所有元素时，会形成一个陪集。

7.The left coset 陪集 of a subgroup can be represented as gH, where g is an element of the group.

子群的左陪集可以表示为gH，其中g是群中的一个元素。

8.Every element in a coset 陪集 has the same order as the subgroup it originates from.

每个在陪集中的元素与它来源的子群具有相同的阶。

9.In modular arithmetic, the set of integers can be partitioned into cosets 陪集 based on a given modulus.

在模算术中，整数集可以根据给定的模数划分为陪集。

作文

In the realm of abstract algebra, the concept of a coset plays a crucial role in understanding group theory. A coset is formed when a subgroup is combined with an element of a group, leading to a new set that maintains certain properties of the original group. To illustrate this, let us consider a group G and a subgroup H. The cosets of H in G can be classified into two types: left cosets and right cosets. A left coset is defined as the set of all products of a fixed element from G and each element of H, while a right coset is defined similarly but with the order reversed.Understanding cosets is essential because they help to partition the group into distinct classes. Each element of G belongs to one and only one coset of H, which means that the collection of all cosets forms a partition of the group. This property leads to the concept of the index of a subgroup, which is the number of distinct cosets of H in G. The index is a significant measure that provides insight into the structure of the group.Moreover, cosets are instrumental in the formulation of Lagrange's theorem, a fundamental theorem in group theory. Lagrange's theorem states that the order of a subgroup H divides the order of the group G. This division can be understood through the relationship between the size of the group and the sizes of its cosets. If we denote the order of G as |G| and the order of H as |H|, then the number of distinct left (or right) cosets of H in G is given by the formula |G|/|H|. This theorem not only highlights the significance of cosets but also illustrates how they connect different aspects of group theory.In practical applications, cosets can be found in various fields such as cryptography, coding theory, and even in solving puzzles like the Rubik's Cube. The ability to manipulate and understand cosets allows mathematicians and computer scientists to develop algorithms that rely on group structures. For instance, in cryptography, the security of certain encryption methods is based on the complexity of group-related problems, where cosets may play a role in defining the underlying mathematical framework.In conclusion, the concept of a coset is foundational in the study of group theory within abstract algebra. It serves as a bridge connecting various elements of group structure, including subgroups, indices, and Lagrange's theorem. By exploring the properties and applications of cosets, one gains a deeper appreciation of the elegance and utility of algebraic structures. As one delves further into the world of mathematics, the understanding of cosets will undoubtedly enhance one's ability to tackle complex problems and appreciate the beauty of mathematical relationships.

在抽象代数的领域中，陪集的概念在理解群论方面起着至关重要的作用。陪集是通过将一个子群与群中的一个元素结合而形成的新集合，这个新集合保持了原始群的一些特性。为了说明这一点，让我们考虑一个群 G 和一个子群 H。H 在 G 中的 陪集 可以分为两种类型：左 陪集 和右 陪集。左 陪集 被定义为来自 G 的一个固定元素与 H 中每个元素的所有乘积的集合，而右 陪集 则以相似的方式定义，但顺序相反。理解 陪集 是必要的，因为它们有助于将群划分为不同的类。G 中的每个元素都属于 H 的一个且仅一个 陪集，这意味着所有 陪集 的集合形成了该群的一个划分。这一属性引出了子群的指数的概念，即 H 在 G 中的不同 陪集 的数量。指数是一个重要的度量，提供了对群结构的洞察。此外，陪集 在拉格朗日定理的表述中也起着重要作用，这是群论中的一个基本定理。拉格朗日定理指出，子群 H 的阶数可以整除群 G 的阶数。这种划分可以通过群的大小与其 陪集 的大小之间的关系来理解。如果我们将 G 的阶数表示为 |G|，将 H 的阶数表示为 |H|，那么 H 在 G 中的不同左（或右）陪集 的数量由公式 |G|/|H| 给出。这个定理不仅突显了 陪集 的重要性，还说明了它们如何连接群论的不同方面。在实际应用中，陪集 可以在密码学、编码理论以及解决像魔方这样的难题等多个领域中找到。操纵和理解 陪集 的能力使数学家和计算机科学家能够开发依赖于群结构的算法。例如，在密码学中，某些加密方法的安全性基于与群相关问题的复杂性，其中 陪集 可能在定义基础数学框架中发挥作用。总之，陪集 的概念在抽象代数的群论研究中是基础性的。它作为连接群结构各个元素的桥梁，包括子群、指数和拉格朗日定理。通过探索 陪集 的性质和应用，人们将更深刻地欣赏代数结构的优雅和实用性。当人们进一步深入数学的世界时，对 陪集 的理解无疑将增强他们解决复杂问题的能力，并欣赏数学关系的美妙。