Abstract Algebra 69: Orbits

Abstract Algebra 69: Orbits

Abstract: We give an introduction to orbits, which are used in the orbit-stabilizer theorem. If G is a group acting on a set S, and if i is an element of S, then the orbit of i is the set of set elements that i can map to under the group action. The stabilizer is always a subset of S.

This video accompanies the class "Introduction to Abstract Algebra" at Colorado State University:

[danielohadiro]

Thank you so much.
please i came across this question could you help me with hint on how to solve it.
Consider the following group actions (X, G). In each case, describe the set
of orbits, and the stabilizer of a chosen point from each orbit.
(1) Let G1 = Sn act on the set X1 = {1, . . ., n} by permutations.
(2) Let G2∼= (R, +) act on the plane X2 = R2 by horizontal translations.
(3) Let X3 be the (boundary of the) unit square in the plane R
2 Let G3 = Sym(X3)
be the set of symmetries of the square.