Math 545 Midterm 2

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Jorge732
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264214
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Math 545 Midterm 2
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2014-02-27 03:08:26
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Midterm 2
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  1. Definition 4.8
    the symmetric group of degree n
    • The group consisting of the set of Sn of
    • all permutations on A={1,2,3,….,n}, under the operation of permutation multiplication
    • is called the symmetric group of degree n.
  2. Definition 4.11
    A permutation of theta-ele- Sn is called a cycle
    if it is
    • A permutation of theta-ele- Sn is called
    • a cycle if it is of the form (a1,a2,a3,…,an). The length of a cycle is the
    • number of elements in it. We call a cycle a k-cycle if there are k elements in
    • it. Two cycles are disjoint if they have no common elements in them.
  3. Definition 4.22
    A Permutation FEE is called an even permutation
    if
    and it is called an odd permutation if
    • A Permutation FEE is called an even
    • permutation if it can be written as a product of even number 2-cycles, and it
    • is called an odd permutation if it can be written as a product of an odd number
    • of 2-cycles.
  4. Definition 5.4
    left coset of H in G
    is called the right coset
    • Let G be a group, H a subgroup of G, and
    • a-element-G. Then the se aH={ah|h-ele-H} is called a left coset of H in G, and
    • the set Ha={ha|h-ele-H} is called the right coset.
  5. Definition 5.10 
    the number of distinct left cosets of H in G is
    called
    • Let G be a finite group and H a subgroup
    • of G. Then the number of distinct left cosets of H in G is called the index of
    • of H in G and is denoted [G:H].
  6. Definition 6.2
    A map FEE: GàG’ from a group G to a group G’ is called
    • A map FEE: GàG’ from a group G to a group G’
    • is called a homomorphism if

    •                        
    • in G  <--   FEE(ab)=FEE(a)FEE(b)  -->       in G’
  7. Definition 6.10
    Let FEE:GàG’ be a homomorphism and let e’ be the identity
    in G’. Then the kernel of FEE is
    • Let FEE:GàG’ be a homomorphism and let e’
    • be the identity in G’. Then the kernel of FEE is the set {x-ele-G|FEE(x)=e’},
    • and denoted ker FEE
  8. Definition 6.16
    A homomorphism FEE:GàG’ that is 1-1 and onto is called
    • A homomorphism FEE:GàG’ that is 1-1 and onto is called
    • an isomorphism. Two groups G and G’ are called isomorphic, written GcongruentG’,
    • if there exists some isomorphism FEE:GàG’

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