Abstract Algebra Key Definitions Theorems

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popolisj
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Abstract Algebra Key Definitions Theorems
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2014-10-09 14:36:53
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Abstract Algebra Theorems
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Review for a Abstract Algebra Test
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  1. State the Well Ordering Principle
    Every nonempty set of positive integers contains a smallest element.
  2. State the Division Algorithm
    Let a and b be integers where b > 0. Then there exists unique integers q&r such that: a = bq + r where 0< r < b
  3. Linear Combination
    For any nonzero integers a and b, there exists integers s & t such that gcd(a,b) = as + bt. Moreover, gcd(a,b) is the smallest positive integer of the form as + bt.
  4. If a and b are relatively prime...
    ... then there exists integers s and t such that as + bt = 1.
  5. Define Euclid's Lemma
    If p is a prime that divides ab, then p divides a or p divides b.
  6. Define the GCD
    • Greatest Common Divisor
    • The GCD of two nonzero integers a and b is the largest of all common divisors of a and b. We denote this integer gcd(a,b).
  7. Define the LCM
    • Least Common Multiple
    • The LCM of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b. We denote this as lcm(a,b).
  8. Define an Equivalence Relation
    • An equivalence relation on a set S is a set of ordered pairs R of S such that 
    • ~(a,a) is an element of R for all a element of S (reflexive)
    • ~(a,b) is an element of R then (b,a) is an element of R (symmetric)
    • ~If (a,b) is an element of R and (b,c) is an element of R then (a,c) is an element of R (Transitive)
  9. Partition of a Set
    A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S.
  10. Define a function
    A function (mapping) is a rule that assigns each a of A a unique element b of B.
  11. Dihedral Group
    a gropu denoted Dn of order 2n
  12. What is a group?
    Let G be a set under a binary operation. A group assigns to each ordered pair (a,b) of elements of G an element in G denoted by ab.
  13. What is a binary operation?
    A binary operation on set G is a function that assigns each ordered pair of elements of G an element of G.
  14. What three properties must a group satisfy?
    • 1. Associativity. i.e. (ab)c = a(bc) for all a,b,c in G.
    • 2. Identity. i.e. there is an element called e such that aeea = a for all a in G.
    • 3. Inverses. i.e. there is an element b such that ab=ba=e.
  15. What does it mean to say a group is Abelian?
    The group has the property that ab = ba for every pair of elements a and b.
  16. What is the Uniqueness of the Identity theorem?
    In a group G, there is only one identity element.
  17. State the Cancellation Theorem
    Right and Left cancellation holds such that ba=ca implies b=c, and ab=ac implies b=c.
  18. State the Uniqueness of Inverses Theorem
    For each element a in a group G there is a unique element b in G such that ab=ba=e.
  19. What is the Socks-Shoes-Property?
    • For group elements a and b,
    • (ab)-1 =b-1a-1
  20. Define the Order of a Group
    The order of a group (finite or infinite) is the number of elements in this group. We denote the order of group G as |G|.
  21. Define the Order of an Element
    Let G be a group and g be an element of G. The order of an element g in G is the smallest possible integer n such that gn=e. If no such n exists then we say g has infinite order. Denoted by |g|.
  22. Define a Subgroup
    If a subset H of a group G is also a group under the same operation of G, we say that H is a subgroup of G. Denoted H<G
  23. What are the steps in the One-Step Subgroup test?
    • 1. Identify the property P that distinguishes the elements of H
    • 2. Prove that the identity has the property P
    • 3. Assume that two elements a and b have property P
    • 4. Use the assumption that a and b have property P to show that ab-1 also has property P
  24. State the One-Step Subgroup test
    • Let G be a group and H a nonempty subset of G.
    • (multiplication) If ab-1 is in H whenever a and b are in H, then H is a subgroup of G.
    • (additive) If a - b is in H whenever a and b are in H, then H is a subgroup of G.
  25. State the Two-Step Subgroup Test
    • Let G be a group and let H be a nonempty subset of G.
    • If ab is in H whenever a and b are in H, and a-1 is in H whenever a is in H, then H is a subgroup of G.
  26. State the Finite-Subgroup Test

    When is this used?
    Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H is a subgroup of G. 

    used when we have finite groups
  27. What is <a> ?
    • Cyclic Group - Let G be a group, and let a be any element of g. Then, <a> is a subgroup denoted by
    • <a> = {an | n is an element of Z}
  28. How do we denote the center of a group G?
    Z(G)
  29. Define the Center of a Group
    • The center of a group G is the subset of elements in G that commute with every element of G. 
    • i.e. Z(G) = {a in G | axxa for all x in G}

    The center of a group G is a subgroup of G.
  30. Define the Centralizer
    • Let a be a fixed element of a group G. The centralizer of a in G is the set of all elements in G that commute with a. Denoted C(a).
    • i.e. C(a) = {g in G | ga = ag}

    The centralizer is a subgroup
  31. What does it mean to say a group is Cyclic?
    • A group is cyclic such that there exists an element a in G such that G = {a| n is an element of Z}
    • denoted <a>
  32. What is a Generator?
    An element a is a generator of G when an = G

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