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cmj1988
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I'm just wondering if there is some sort of relationship between right coset and orbit of x. We just got to cosets, and it seems like the properties of cosets are eerily similar to orbits.
You might be interested in theorem 3 on page 4 of this article.cmj1988 said:I'm just wondering if there is some sort of relationship between right coset and orbit of x. We just got to cosets, and it seems like the properties of cosets are eerily similar to orbits.
The study of group actions thus divides neatly into two problems: the internal problem of understanding the action within single orbits (equivalent to studying the canonical action in coset spaces) and the external problem of understanding how the orbits are put together to form the set X.
A right coset is a subset of a group formed by multiplying each element of the subgroup by a specific element of the group. This results in a set of elements that are all equivalent to each other, but may be in a different order.
A right coset is formed by multiplying the elements of the subgroup on the right, while a left coset is formed by multiplying the elements on the left. This can result in different elements being included in the coset, but the overall structure and properties remain the same.
Cosets allow us to partition a group into smaller, equivalent subsets. This can help us to understand the structure and properties of a group, as well as identify relationships between elements and subgroups.
Orbits are a special type of right coset, where the group is acting on itself through a specific operation. The orbit contains all elements that can be reached from a given element in the group by applying the operation. This is similar to how a right coset contains all elements that can be obtained by multiplying the subgroup by a specific element.
Yes, right cosets can be used to identify subgroups of a group. If a subgroup is formed by multiplying a specific element of the group by all elements of the subgroup, then the right coset of that element will be equal to the subgroup. This can help us to identify and understand the structure of a group and its subgroups.