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2013-06-27 15:32:45
Math Algebra

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