answer:a) i. A square represents the dihedral group D_8, with 8 symmetries including rotations by 90°, 180°, and 270° and reflections across four axes. ii. A rectangle or diamond represents the dihedral group D_4, with 8 symmetries including rotations by 90°, 180°, and 270° and reflections across two axes. iii. A regular pentagon exemplifies the cyclic group C_5, with 5 symmetries corresponding to rotations by multiples of 72°. iv. A regular hexagon represents the cyclic group C_6, with 6 symmetries corresponding to rotations by multiples of 60°. b) No, there does not exist a shape in the plane with the symmetry group S_4. A distance-preserving symmetry in the plane is necessarily an affine transformation. If a subgroup of order 3 has a fixed point, then all three elements of that subgroup share the same fixed point, which would require three points to be equidistant from each other, implying an equilateral triangle. However, S_4 has eight elements of order 3, which cannot be arranged in such a way that they share fixed points without violating the planar nature of the symmetry. Additionally, distinct elements of S_4 must act in distinct ways on the shape, which is not possible with a planar figure. In 3D, a regular tetrahedron has a symmetry group isomorphic to A_4, a subgroup of S_4, but this is not a planar figure.

answer:Glycogen is a complex polysaccharide that serves as an energy reserve in heterotrophic organisms. Primarily found in the liver and muscles of animals, glycogen is also utilized by certain fungi and bacteria. When energy is needed, glycogen is broken down through the action of the enzyme glycogen phosphorylase, which cleaves it into glucose 1-phosphate monomers. These monomers can then be converted into glucose-6-phosphate, entering the glycolysis pathway to generate ATP for cellular energy.

answer:For patients over 40 years old with aplastic anemia, the treatment of choice often includes immunosuppressive therapy with drugs like cyclosporine and anti-thymocyte globulin (ATG). These medications help stimulate the bone marrow to produce new blood cells. In some cases, a stem cell transplant may also be considered, but this depends on various factors, including the patient's overall health and availability of a suitable donor.

answer:The individual who can deduct education expenses as work-related is Option C, the corporate executive attending an executive MBA program. The costs of an executive MBA are generally deductible because they are related to the executive's job and help enhance their skills in the corporate environment. Explanation of other options: a. The accounting degree is not directly related to the work of a real estate broker, making option A incorrect for tax deduction purposes. b. The building contractor's license is not relevant to the job of a CPA, so option B's expenses would not qualify as a work-related educational deduction. d. Although an accounting bookkeeper may aspire to become a CPA, the CPA review course is not a direct requirement for their current job as a bookkeeper, making option D's expenses nondeductible as work-related education.