,1. Let G be the dihedral group of symmetries of a square. is the action of G on the vertices a faithful action? on the diagonals?
=== Does for every vertex in the square there exist an element of G that moves that vertex? Same question for diagonals. If you kn wo how D acts on the square this musn't be big problem
2. A group G operates faithfully on a set S of five elements, and there are two orbits, one of order 3 and one of order 2. what are the possibilities for G?
=== Do you know that the size of orbits always divides the order of the group?
can anyone please help me to solve these problems?