1. What is the one step subgroup test?
    Let G be a group and H be a non-empty subset of G. Let Image Upload 1, if Image Upload 2 then H is a subgroup.
  2. What is the two-step subgroup test?
    Let G be a group and H a subgroup of G. Let Image Upload 3. If H is closed under inverses and closed under the group operation H is a subgroup.
  3. Define: The center of the group G.
    The center of a group G, denoted Image Upload 4.
  4. Define: Centralizer of a in G.
    The centralizer of Image Upload 5Image Upload 6.
  5. Let G be a group, with Image Upload 7 such that Image Upload 8. What can be said about i and j.
    If G is infinite i=j. If G is finite, i-j divides n.
  6. What is the order of Image Upload 9 and Image Upload 10.
    Image Upload 11 and Image Upload 12.
  7. Define: Coset of H in G.
    Let Image Upload 13 be a subgroup of Image Upload 14. Let Image Upload 15, the coset of H in G is defined as Image Upload 16.
  8. State Lagranges Theorem.
    If H is a subgroup of the finite group G. Then Image Upload 17 divides Image Upload 18.
  9. Define: Image Upload 19.
    Image Upload 20 is the order of Image Upload 21.
  10. How many groups of order Image Upload 22 are they. Where Image Upload 23 is prime.
    1 up to isomorphism, Image Upload 24.
  11. What is Fermat's Little Theorem?
    Let Image Upload 25 be prime. Then for all Image Upload 26.
  12. How many groups of order 2p are there? Where  p is a prime greater than 2
    two groups, Image Upload 27 or Image Upload 28.
  13. Define: Let G be a group and Image Upload 29 be a set. Let Image Upload 30. Define the stabilizer of Image Upload 31 in G.
    The stabilizer of Image Upload 32 in G denoted Image Upload 33
  14. Define: Let Image Upload 34 be a set and Image Upload 35 be a group acting on Image Upload 36. Define the orbit of the point Image Upload 37.
    The orbit of Image Upload 38 in Image Upload 39, denoted Image Upload 40.
  15. What is the orbit-stabilizer theorem.
    for any Image Upload 41Image Upload 42.
  16. Let G and H be finite cyclic groups. When is Image Upload 43 cylic
    If and only if order of G and H are relatively prime.
  17. Define: Image Upload 44.
    Image Upload 45
  18. Define: A characteristic subgroup of G.
    N is a characteristic subgroup of G if Image Upload 46.
  19. Define: Normal subgroup of G.
    Let Image Upload 47 be a subgroup of the group Image Upload 48. Then N is normal if and only if Image Upload 49. Denoted Image Upload 50.
  20. What is the normal subgroup test?
    If Image Upload 51 and Image Upload 52 then N is normal in G.
  21. Let G be a group and Image Upload 53 be the center of G. Assume Image Upload 54 is cyclic, what can be said about G?
    G is Abelian.
  22. How many groups of order Image Upload 55 are there?
    two, Image Upload 56 or Image Upload 57.
  23. State the first isomorphism Theorem.
    Let Image Upload 58 be an onto homomorphism of groups, rings or modules. Then Image Upload 59.
  24. What is the second isomorphism Theorem for groups.
    If K is a subgroup of G and H is a normal subgroup of G, then Image Upload 60.
  25. State the Third Isomorphism Theorem.
    If M and N are normal subgroups of G and N is a subgroup of M, Image Upload 61.
  26. Define: Integral Domain.
    An integral domain is a commutative ring with unity and no zero divisors.
  27. Define: Characteristic of a Ring
    The least positive integer Image Upload 62 such that Image Upload 63 for all Image Upload 64. If no such integer exists, the characteristic is said to be zero.
  28. What is the characteristic of an integral domain
    either zero or prime.
  29. Define: Ideal of a ring.
    A subset of R is an ideal if it is a subring and has the property that for all Image Upload 65 and Image Upload 66.
  30. Test that I is an ideal of R
    • 1) Check that I is nonempty.
    • 2) closed under addition. 
    • 3) absorbs elements from R.
  31. Define: Prime and maximal ideals.
    An ideal Image Upload 67 is prime if Image Upload 68 is multiplicative closed. And ideal M of R is maximal if the only ideal containing M is R.
  32. Let R be commutative. When is Image Upload 69 and integral domain?
    If and only if A is prime.
  33. Let R be commutative. When is Image Upload 70 a field?
    If and only if A maximal.
  34. What is the chinese remainder theorem?
    Let R be a ring and I and J be coprime ideals of R, then Image Upload 71.
  35. What is the mod p test for irreducibility.
    Let f(x) be a polynomial over the integers of degree greater than one. If there exists p, prime such that Image Upload 72 is irreducible and doesn't change degree then f(x) is irreducible.
  36. What is Eisenstein's Criterion
    Let Image Upload 73, if there exists a prime p such that Image Upload 74Image Upload 75 and Image Upload 76, then f(x) is irreducible.
  37. When does prime imply irreducible
    In an integral domain.
  38. In a UFD is irreducible prime.
  39. Let f(x) be a polynomial over the field F. When does f(x) have a multiple zero in an extension E.
    If and only if f(x) and f'(x) have a common factor in F[x].
  40. Let f(x) be an irreducible polynomial over K. How many multiple zeros does f(x) have?
    If K has characteristic 0, then f(x) has no multiple zeros. If K has characteristic p then f has a multiple zero if it is of the form Image Upload 77 for some g(x) in K[x].
  41. Define: Perfect Field.
    A field K is called perfect if it has characteristic zero or the map Image Upload 78 defined by Image Upload 79 is onto.
  42. Let f(x) be an irreducible polynomial over F and E be the splitting field of f(x). What can be said about the multiplicity of the zeros in E.
    Every zero has the same multiplicity.
  43. Define the minimal polynomial for a over F.
    The minimal polynomial for a over F is the monic polynomial of least degree that has a has a root.
  44. State the tower law.
    Let F be a field and E be a finite extension of F. Let K be a finite extension of E. Then K is a finite extension of F and Image Upload 80.
  45. State the primitive element theorem.
    If F has characteristic zero and with a and b algebraic over F then there exists c such that F(a,b)=F(c)
  46. Define: Conjugacy class of a.
    Image Upload 81.
  47. Let G be a finite group. How many conjugates does a have in G?
    Image Upload 82
  48. What is the class equation?
    For any finite group G, Image Upload 83.
  49. What is Sylows First Theorem?
    Let G be a finite group and let p be a prime. If Image Upload 84 divides the order of G then G has at least one subgroup of order Image Upload 85.
  50. Define: Sylow p-subgroup of the finite group G.
    Let p be prime and let k be the largest power of p that divides the order of G. Then any subgroup of order Image Upload 86 is a sylow p-subgroup of G.
  51. What does it mean for two subgroups to be conjugate?
    H and K are conjugate if there exists x in G such that Image Upload 87.
  52. What is Sylow's second theorem
    If H is a subgroup of the finite group G, and the order of H is a power of a prime divisor of G. Then H is contained in some sylow p-subgroup of G
  53. What is Sylow's Third Theorem?
    • Let p be a prime and let Image Upload 88 where p does not divide m. Then Image Upload 89 the number of sylow p-subgroups of G satisfies the follow properties 
    • 1) Image Upload 90
    • 2) Image Upload 91
    • Futhermore all sylow p-subgrops are conjugate.
  54. What is the second Isomorphism Theorem for Rings.
    If A and B are ideals of R then Image Upload 92.
Card Set
Definitions from groups, rings and fields