Relationship between orbits and cosets

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SUMMARY

The relationship between orbits and cosets is defined by group actions, where a group permutes cosets through group multiplication. An orbit consists of all points reachable from a specific point under the group action. While all orbits are not necessarily cosets, if a group acts transitively on a set, the stabilizer of any point forms a subgroup, and the coset space becomes isomorphic to the original set as a G-space. This relationship highlights the significance of G-sets and G-equivariant sets in understanding group actions.

PREREQUISITES
  • Understanding of group actions in abstract algebra
  • Familiarity with the concepts of orbits and cosets
  • Knowledge of stabilizer subgroups
  • Basic comprehension of G-sets and G-equivariant sets
NEXT STEPS
  • Study the properties of G-sets and their applications
  • Explore the concept of transitive group actions in detail
  • Learn about the isomorphism between coset spaces and G-spaces
  • Investigate examples of group actions on various mathematical structures
USEFUL FOR

Mathematicians, particularly those specializing in group theory, abstract algebra students, and researchers exploring the applications of group actions in various mathematical contexts.

potmobius
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How are orbits and cosets related? Are all orbits cosets? Are all cosets orbits? Also, what exactly are G-sets and G-equivariant sets?
 
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potmobius said:
How are orbits and cosets related? Are all orbits cosets? Are all cosets orbits? Also, what exactly are G-sets and G-equivariant sets?

A group permutes cosets by group multiplication.

An orbit of a point in a group action is just all of the points that the point reaches under the action.

But groups can act on may things and they certainly do not have to be cosets.

However, suppose that a group acts transitively on a set. (This means that there is only one orbit in the group action.)

Then the stabilizer of any point is a subgroup of the group (can you prove that?) and the coset space is isomorphic to the original set as a G-space. this you should prove for yourself.

So the group action can be represented as the action on the set of cosets of the stabilizer subgroup.
 
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