Final answer:
The stabilizer of a vertex in the dihedral group D4 is the cyclic group of order 2. The stabilizer of an edge in D4 is also the cyclic group of order 2.
Step-by-step explanation:
(a) The stabilizer of a vertex in the dihedral group D4 consists of the symmetries that leave that vertex fixed. In other words, it is the subgroup of D4 that preserves the position of the vertex. The stabilizer of a vertex is isomorphic to the cyclic group of order 2, denoted as C2. This is because there are two symmetries that fix a vertex: the identity and the reflection about the axis passing through the vertex.
(b) The stabilizer of an edge in D4 consists of the symmetries that leave that edge fixed. In other words, it is the subgroup of D4 that preserves the position of the edge. The stabilizer of an edge is isomorphic to the cyclic group of order 2, denoted as C2. This is because there are two symmetries that fix an edge: the identity and the rotation by 180 degrees about the center of the square.