Extremely understandble explanation of the group actions here...Thank you very much for putting such effort into this. I finally understand the cocenpt of group actions and also the connection to the Cayleys Theorem.
Actions are the essence of scalar multiplication. For example, a vector space can be described as a group which has a field acting on it. The field forms the "scalars" and the group forms the "vectors". Sometimes the intuition coming from "scaling" is more useful than the intuition coming from permuting.
Group actions are dual to permutation representations! Injection is dual to surjection creates bijection or isomorphism. Points are dual to lines -- the principle of duality in geometry.
Extremely understandble explanation of the group actions here...Thank you very much for putting such effort into this. I finally understand the cocenpt of group actions and also the connection to the Cayleys Theorem.
set permutations its just better specially with the venn function diagram representations with arrows.
Good Work. Thank you. This nice explanation of group action should have at least one non-trivial example.
What about structures for different kernals/quotient groups? What about typical homorphisms into set A?
Group actions are dual to permutation representations!
Injection is dual to surjection creates bijection or isomorphism.
Why is there a preference for using the group action method rather than the more intuitive permutation representations?
Actions are the essence of scalar multiplication. For example, a vector space can be described as a group which has a field acting on it. The field forms the "scalars" and the group forms the "vectors". Sometimes the intuition coming from "scaling" is more useful than the intuition coming from permuting.
Group actions are dual to permutation representations!
Injection is dual to surjection creates bijection or isomorphism.
Points are dual to lines -- the principle of duality in geometry.
Can one imagine S_A as the set of all bijections from A to itself, even if A isnt finite?
Yes. It makes perfect sense, even for infinite sets A.