The document discusses group actions and permutations. It defines a group action as a function from a group G to a nonempty set M. This function assigns each group element g in G a permutation of M. Group actions make precise the idea that group elements act on or induce movements of elements in the set M, like the permutations of vertices of a square induced by the symmetries of the square.

1. 2. In the following problems, Sn is the group of permutations of {1 2, ... , n} under function
composition. (c) Find permutations sigma and T satisfying gcd(o(sigma T), o(sigma)) =
gcd(o((sigma T), o(T)) = 1. use a is not equal to e Tau is not equal to e
Solution
the symmetry group of a square, the permutations "describe" the movement of the vertices of
the square induced by the group of symmetries. It is common to say that these group elements
are "acting" on the set of vertices of the square. This idea can be made precise by formally
defining a group action.[12]
Let G be a group and M a nonempty set. An action of G on M is a function f: G