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노트

위키데이터

• ID : Q288465

말뭉치

1. Group action will often take place when social agents realize they are more likely to achieve their goal when acting together rather than individually.
2. In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space.
3. A group action on a (finite-dimensional) vector space is called a representation of the group.
4. In other words, in a faithful group action, different elements of G induce different permutations of X .
5. In other words, in a faithful group action, different elements of induce different permutations of .
6. In a group action, a group permutes the elements of .
7. For a given , the set , where the group action moves , is called the group orbit of .
8. Historically, the first group action studied was the action of the Galois group on the roots of a polynomial.
9. Section 7 uses a group action by automorphisms to define the semidirect product of two groups.
10. There are a few questions that come up when encountering a new group action.
11. The first condition for a group action holds by associativity of the group, and the second condition follows from the definition of the identity element.
12. In other words, an equivariant map is a homomorphism with respect to the group action; it is therefore also sometimes called a G-map or G-homomorphism.
13. Thus a Lie group action is defined to be a smooth homomorphism from a Lie group \({G}\) to \({\textrm{Diff}(M)}\), the Lie group of diffeomorphisms of a manifold \({M}\).
14. Note that this function must implement a group action from the right.
15. OrbitStabilizerAlgorithm performs an orbit stabilizer algorithm for the group G acting with the generators gens via the generator images gens and the group action act on the element pnt .
16. Mathematically, an external set is the set Ω, which is endowed with the action of a group G via the group action μ.
17. We can formalize this notion with the concept of a group action.
18. Now since the symmetric group is the group of all bijections we can think of a group action as a homomorphism from to .
19. This definition allows us to easily study the concept of a group action in the framework of category theory.
20. A group action can be run by a single firm of solicitors acting on behalf of all the individuals, or those individuals could be represented by a number of different firms.
21. The term group action or action of a group is used for the notion defined here.
22. A group action is termed faithful if no non-identity element of the group fixes everything.