What is the definition and purpose of a Group Action in mathematics?

Group action is a fundamental concept in mathematics that arises in various fields such as algebra, geometry, and graph theory. It is a mathematical operation that describes how a group (a set of objects with a defined binary operation) acts on another set. In simple terms, it is a way of studying symmetries and transformations of objects by using the structure of a group. The purpose of group action is to provide a powerful tool for understanding and analyzing the behavior of mathematical objects and their relationships. In this introduction, we will delve deeper into the definition and purpose of group action in mathematics.

In sociology, a group action is a situation in which a large number of agents take action simultaneously in order to achieve a common goal; their actions are usually coordinated.

Group action will often take place when social agents realise they are more likely to achieve their goal when acting together rather than individually. Group action differs from group behaviours, which are uncoordinated, and also from mass actions, which are more limited in place.

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