Final answer:
In the dihedral group D₄, cyclic subgroups are formed by rotations and include the identity and the powers of that rotation. A non-cyclic subgroup that contains rho or rho³ is the group D₄ itself, as it also includes the identity, all rotations, and reflections. Non-cyclic subgroups other than D₄ consist of the identity element and each individual reflection.
Step-by-step explanation:
The group D₄, also known as the dihedral group of order 8, represents the symmetries of a square, including rotations and reflections. The question asks for finding cyclic subgroups, a non-cyclic subgroup containing a specific rotation, and all non-cyclic subgroups of D₄.
Part (a): Cyclic Subgroups of D₄
Cyclic subgroups are generated by a single element and contain all powers of that element. In D₄, the rotations rho, with orders 1, 2, and 4, will form cyclic subgroups {e}, {e, rho²}, and {e, rho, rho², rho³} respectively, where e is the identity element. Each reflection does not form a cyclic subgroup as it is of order 2 and only pairs with the identity.
Part (b): Non-Cyclic Subgroup Containing rho or rho³
If a non-cyclic subgroup contains the rotation rho or rho³, it must also contain all powers of that element. Since rho³ = rho⁻¹, any subgroup with rho or rho³ will also contain rho² and the identity e. It should also contain all reflections since they can be expressed as a reflection combined with a rotation (for example, sigma * rho) which must also be in the subgroup if it contains all rotations. As a result, such a subgroup would actually contain all elements of D₄, thus the subgroup is D₄ itself.
Part (c): All Non-Cyclic Subgroups of D₄
The non-cyclic subgroups of D₄, other than D₄ itself, consist of the subgroups that contain a reflection and the identity. There are 4 such subgroups corresponding to the 4 reflections of the square, each subgroup containing the reflection and the identity element {e, sigma₁}, {e, sigma₂}, {e, sigma₃}, and {e, sigma₄}.