Algebra

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  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, if Image Upload 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. 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. Define: Centralizer of a in G.
    The centralizer of Image UploadImage Upload.
  5. Let G be a group, with Image Upload such that Image Upload. 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 and Image Upload.
    Image Upload and Image Upload.
  7. Define: Coset of H in G.
    Let Image Upload be a subgroup of Image Upload. Let Image Upload, the coset of H in G is defined as Image Upload.
  8. State Lagranges Theorem.
    If H is a subgroup of the finite group G. Then Image Upload divides Image Upload.
  9. Define: Image Upload.
    Image Upload is the order of Image Upload.
  10. How many groups of order Image Upload are they. Where Image Upload is prime.
    1 up to isomorphism, Image Upload.
  11. What is Fermat's Little Theorem?
    Let Image Upload be prime. Then for all Image Upload.
  12. How many groups of order 2p are there? Where  p is a prime greater than 2
    two groups, Image Upload or Image Upload.
  13. Define: Let G be a group and Image Upload be a set. Let Image Upload. Define the stabilizer of Image Upload in G.
    The stabilizer of Image Upload in G denoted Image Upload
  14. Define: Let Image Upload be a set and Image Upload be a group acting on Image Upload. Define the orbit of the point Image Upload.
    The orbit of Image Upload in Image Upload, denoted Image Upload.
  15. What is the orbit-stabilizer theorem.
    for any Image UploadImage Upload.
  16. Let G and H be finite cyclic groups. When is Image Upload cylic
    If and only if order of G and H are relatively prime.
  17. Define: Image Upload.
    Image Upload
  18. Define: A characteristic subgroup of G.
    N is a characteristic subgroup of G if Image Upload.
  19. Define: Normal subgroup of G.
    Let Image Upload be a subgroup of the group Image Upload. Then N is normal if and only if Image Upload. Denoted Image Upload.
  20. What is the normal subgroup test?
    If Image Upload and Image Upload then N is normal in G.
  21. Let G be a group and Image Upload be the center of G. Assume Image Upload is cyclic, what can be said about G?
    G is Abelian.
  22. How many groups of order Image Upload are there?
    two, Image Upload or Image Upload.
  23. State the first isomorphism Theorem.
    Let Image Upload be an onto homomorphism of groups, rings or modules. Then Image Upload.
  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.
  25. State the Third Isomorphism Theorem.
    If M and N are normal subgroups of G and N is a subgroup of M, Image Upload.
  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 such that Image Upload for all Image Upload. 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 and Image Upload.
  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 is prime if Image Upload 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 and integral domain?
    If and only if A is prime.
  33. Let R be commutative. When is Image Upload 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.
  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 is irreducible and doesn't change degree then f(x) is irreducible.
  36. What is Eisenstein's Criterion
    Let Image Upload, if there exists a prime p such that Image UploadImage Upload and Image Upload, then f(x) is irreducible.
  37. When does prime imply irreducible
    In an integral domain.
  38. In a UFD is irreducible prime.
    Yes.
  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 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 defined by Image Upload 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.
  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.
  47. Let G be a finite group. How many conjugates does a have in G?
    Image Upload
  48. What is the class equation?
    For any finite group G, Image Upload.
  49. What is Sylows First Theorem?
    Let G be a finite group and let p be a prime. If Image Upload divides the order of G then G has at least one subgroup of order Image Upload.
  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 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.
  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 where p does not divide m. Then Image Upload the number of sylow p-subgroups of G satisfies the follow properties 
    • 1) Image Upload
    • 2) Image Upload
    • 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.
Author:
NhanNguyen
ID:
225174
Card Set:
Algebra
Updated:
2013-06-27 19:32:45
Tags:
Math Algebra
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Description:
Definitions from groups, rings and fields
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