Final answer:
The question involves proving that a subgroup H is a normal subgroup of the dihedral group D10 and that the quotient group D10/H is isomorphic to D5. The subgroup H generated by the element x5 in D10 is proven to be normal by showing the equivalence of conjugate subsets, and the isomorphism is established by demonstrating a scaling down of symmetry operations and a bijective mapping that preserves group operations.
Step-by-step explanation:
The question pertains to group theory, a topic within abstract algebra in mathematics. It involves the dihedral group D10, which is the group of symmetries of a regular decagon, including rotations and reflections. The student is asked to prove that a certain subgroup H is a normal subgroup of D10 and that the quotient group D10/H is isomorphic to D5, the group of symmetries of a regular pentagon.
To prove that H is normal, we must show that for every element g in D10, the conjugate gHg-1 is equal to H. Given that H is a subgroup generated by the element x5, which corresponds to a 72-degree rotation in D10, all conjugates of x5 by other elements of D10 will still be rotations by multiples of 72 degrees, thus staying within H. This shows that H is indeed normal in D10.
For the isomorphism between D10/H and D5, notice that D10 consists of 10 elements - 5 rotations and 5 reflections while D5 consists of 5 elements - rotations only. Since x5 is a 72-degree rotation, factoring out the subgroup H generated by x5 effectively 'scales down' the symmetry operations of D10 by a factor of 2. Specifically, each corresponding element in D10 factoring by H maps to an element in D5. This forms a bijection, preserving group operations, and thus D10/H is isomorphic to D5.