Prove that there is no simple group of order 210 = 2 - 3 - 5 - 7. Prove that there is no simple group of order 280 = 23 - 5 - '7. Prove that there is no simple group of order 216 = 23 - 33. Prove that there is no simple group of order 300 = 22 - 3 - 52. Prove that there is no simple group of order 525 = 3 - 52 - '7. Prove that there is no simple group of order 540 = 22 - 33 - 5. . Prove that there is no simple group of order 528 = 24 - 3 - 11. 8. Prove that there is no simple group of order 315 = 32 ' 5 - 7. 9. Prove that there is no simple group of order 396 = 22 - 32 - 11. 10. Prove that there is no simple group of order n, where 201 E n 5 235 and n is not prime. 11. Without using the Generalized Cayley Theorem or its corollaries, prove that there is no simple group of order 112. 12. Without using the 2 - Odd Test, prove that there is no simple group of order 210. 13. You may have noticed that all the \"hard integers\" are even. Choose three odd integers between 200 and 1000. Show that none of these is the order of a simple group unless it is prime. 14. Show that there is no simple group of order pqr, where p, q, and r are primes (p, q, and r need not be distinct). 15. Show that A5 does not contain a subgroup of order 30, 20, or 15. 16. Show that 55 does not contain a subgroup of order 40 or 30. I7. Prove that there is no simple group of order 120 = 23 - 3 - 5. (This exercise is referred to in this chapter.) 18. Prove that if G is a nite group and H is a proper normal subgroup of largest order, then G/H is simple. 19. Suppose that H is a subgroup of a nite group G and that IHI and (IG:H| 1)! are relatively prime. Prove that H is normal in G. What does this tell you about a subgroup of index 2 in a nite group? 20. Suppose that p is the smallest prime that divides IGI. Show that