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In this section we define groupsGroup. Basically every monoidMonoid is also a semigroup. Every semigroup is a groupoid. Now, we consider a monoidMonoid with one more property. So, a groupGroup is a setSet with a binaryBinary operation \(*\) defined on the elements of the setSet. In monoidsMonoid and semigroups also, an operation is applied on any two elements of the setSet S and the nature of the output is studied in S itself. For example for any \(a, b \in S\) , \(a * b \in S\) is known as closureClosure property. GroupsGroup of finite order are useful in coding theory, and cryptography.

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Polya’s Theory

  • 章节

  • 发表日期：

    2025年

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  • R. Rama

摘要

In this section we define groupsGroup. Basically every monoidMonoid is also a semigroup. Every semigroup is a groupoid. Now, we consider a monoidMonoid with one more property. So, a groupGroup is a setSet with a binaryBinary operation \(*\) defined on the elements of the setSet. In monoidsMonoid and semigroups also, an operation is applied on any two elements of the setSet S and the nature of the output is studied in S itself. For example for any \(a, b \in S\) , \(a * b \in S\) is known as closureClosure property. GroupsGroup of finite order are useful in coding theory, and cryptography.