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xiaoxiaoyu
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All the perutations of elements in R(3)(three dimension euclidean space) form a permutation group. This group must be E(3)(euclidean group)?
A permutation group is a mathematical concept that describes a collection of objects that can be rearranged or re-ordered in different ways. In other words, it is a set of transformations that can be applied to a given set of objects in order to change their order or arrangement.
A euclidean group is a type of permutation group that deals specifically with transformations of objects in a Euclidean space, which is a type of mathematical space that follows the principles of Euclidean geometry. In this type of group, the transformations preserve the distances and angles between the objects.
Permutation groups can be considered as a subset of euclidean groups, as they are a type of transformation that preserves the structure of the objects being transformed. However, euclidean groups also include other types of transformations, such as rotations and reflections, while permutation groups only deal with reordering.
Euclidean groups have applications in various fields of science, such as physics, chemistry, and computer science. They are used to describe the symmetries and transformations of physical systems, as well as in the development of algorithms for computer graphics and image processing.
Euclidean groups can be represented using matrices or by using the language of group theory, which involves defining a set of group elements and operations that preserve the properties of the group. They can also be represented using geometric objects, such as points, lines, and rotations, in a Euclidean space.