Math 3121 Abstract Algebra I

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Math 3121Abstract Algebra I

• Lecture 9
• Finish Section 10
• Section 11

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HW Due Today

• Hand in
• Due Tues, Oct 28Page 73 12, 14, 16Page 84
  18
• Pages 94-95 10, 24, 36
• Do not hand in
• Pages 94-95 19, 39 

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Section 10

• Section 10 Cosets and the Theorem of Lagrange
• Modular relations for a subgroup
• Definition Coset
• Theorem of Lagrange For finite groups, the order
  of subgroup divides the order of the group.
• Theorem For finite groups, the order of any
  element divides the order of the group

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Modulo a Subgroup

• Definition Let H be a subgroup of a group G.
  Define relations L and R by
• x L y ? x-1 y in H
• x R y ? x y-1 in H
• We will show that L and R are equivalence
  relations on G.
• We call L left modulo H.
• We call R right modulo H.
• Note
• x L y ? x-1 y h, for some h in H
• ? y x h, for some h in H
• x R y ? x y-1 h, for some h in H
• ? x h y, for some h in H

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Modulo a Subgroup is an Equivalence Relation

• Theorem Let H be a subgroup of a group G. The
  relations L and R defined by
• x L y ? x-1 y in H
• x R y ? x y-1 in H
• are equivalence relations on G.
• Proof We show the three properties for
  equivalence relations
• 1) Reflexive x-1 x e is in H. Thus x L x.
• 2) Symmetric x L y ?x-1 y in H
• ? (x-1 y) -1 in H
• ? y-1 x in H
• ? y L x
• 3) Transitive x L y and y L z ? x-1 y in H
  and y-1 z in H
• ? (x-1 y )( y-1 z) in H
• ? (x-1 z) in H
• ? x L z
• Similarly, for x R y .

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Cosets

• The equivalence classes for these equivalence
  relations are called left and right cosets modulo
  the subgroup.
• Recall x L y ? x-1 y h, for some h in H
• ? y x h, for some h in H
• Cosets are defined as follows
• Definition Let H be a subgroup of a group G.
• The subset
• a H a h h in H
• is called the left coset of H containing a, and
  the subset
• H a a h h in H
• is called the right coset of H containing a.

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Examples

• Cosets of nZ in Z are
• nZ, nZ1, nZ2, , nZ (n-1)
• For example 2Z in Z has two cosets.
• Cosets of 0, 3 in Z6
• Cosets of 0, 2, 4 in Z6

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Abelian versus Nonabelian

• Note In abelian groups the left cosets are the
  right cosets. In nonabelian case left and right
  dont always agree.
• H e, µ1 in G S3 has different left and
  right cosets. For left cosets, make a
  multiplication table GH. For right cosets, make
  a multiplication table HG. (See slides after
  this one).
• H ?0, ?1, ?2 in S3 has same left and right
  cosets.

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Multiplication Table for S3 in Cycle Notation
e (1 2 3) (1 3 2) (2 3) (1 3) (1 2)
e e (1 2 3) ( 13 2) (2 3) (1 3) (1 2)
(1 2 3) (1 2 3) (1 3 2) e (1 2) (2 3) (1 3)
(1 3 2) (1 3 2) e (1 2 3) (1 3) (1 2) (2 3)
(2 3) (2 3) (1 3) (1 2) e (1 2 3) (1 3 2)
(1 3) (1 3) (1 2) (2 3) (1 3 2) e (1 2 3)
(1 2) (1 2) (2 3) (1 3) (1 2 3) (1 3 2) e
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Left Cosets for S3 in Cycle Notation
e (2 3)
e e (2 3)
(1 2 3) (1 2 3) (1 2)
(1 3 2) (1 3 2) (1 3)
(2 3) (2 3) e
(1 3) (1 3) (1 3 2)
(1 2) (1 2) (1 2 3)
Distinct left cosets of H e, (2 3) e, (2
3) e H (2 3) H (1 2 3), (1 2) (1 2 3) H
(1 2) H (1 3 2), (1 3) (1 3 2) H (1 3) H
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Right Cosets of lt(2 3)gt for S3 in Cycle Notation
e (1 2 3) (1 3 2) (2 3) (1 3) (1 2)
e e (1 2 3) ( 13 2) (2 3) (1 3) (1 2)
(2 3) (2 3) (1 3) (1 2) e (1 2 3) (1 3 2)
Distinct right cosets of H e, (2 3) e, (2
3) H H (2 3) (1 2 3), (1 3) H (1 2 3)
H (1 3) (1 3 2), (1 2) H (1 3 2) H (1 2)
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Counting Cosets

• Theorem For a given subgroup of a group, every
  coset has exactly the same number of elements,
  namely the order of the subgroup.
• Proof Let H be a subgroup of a group G. Recall
  the definitions of the cosets aH and Ha.
• a H a h h in H
• H a a h h in H
• Define a map La from H to aH by the formula
  La(g) a g. This is 1-1 and onto.
• Define a map Ra from H to Ha by the formula
  Ra(g) g a. This is 1-1 and onto.

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Lagrange

• Theorem (Lagrange) Let H be a subgroup of a
  finite group G. Then the order of H divides the
  order of G.
• Proof Let n number of left cosets of H, and
  let m the number of elements in H. Then m is
  the number of elements in any left coset. Thus
  n m the number of elements of G. Here m is the
  order of H, and n m is the order of G.

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Orders of Cycles

• The order of an element in a finite group is the
  order of the cyclic group it generates. Thus the
  order of any element divides the order of the
  group.

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HW Section 10

• Dont hand in
• Pages 101 3, 6, 9, 15
• Hand in Tues, Nov 4
• Pages 101-102 6, 8, 10, 36, 40

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Section 11 (as time permits)

• Direct Products and Finitely Generated Abelian
  Groups
• Cartesian Product of sets
• Direct product of groups
• Structure of Zn?Zm
• Structure of products of cyclic groups
• Next time Structure of Finitely Generated
  Abelian Groups

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Cartesian Products

• Definition The Cartesian product of a finite
  collection of sets Sk, for k 1 to n is the set
  of all n-tuples (s1, s2, , sn), with sk in Sk.
  The Cartesian product is denoted by
• S1 S2 Sn
• or by product notations such as

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Projection Maps

• The map pi S1 S2 Sn ? Si that takes the
  n-tuple s (s1, s2, , sn) to its ith component
  si is called the ith projection map.
• In other words For any s in S1 S2 Sn the
  result of applying the ith projection map to s is
  called the ith component of s and denoted by si .
• Note Two elements a and b of the product are
  equal if and only if ai bi, for all i 1,,n.

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Direct Product of Groups

• Theorem Let G1, G2, , Gn be groups with
  multiplicative notation. Define a binary
  operation on the Cartesian product G1G2 Gn by
  (a1, a2, , an)(b1, b2, , bn) (a1 b1, a2 b2,
  , an bn), then G1G2 Gn is a group with this
  binary operation. The set G1G2 Gn with this
  binary operation is called the direct product of
  the groups G1, G2, , Gn.
• Proof Note that the binary operation is defined
  component-wise. That is, if a and b are in the
  product, then (a b)i ai bi. 1) Associativity
  follows because each component binary operation
  is associative. 2) The identity is e the n-tuple
  e (e1, e2, , en), where each ei is the
  identity of its own component group Gi. 3) For
  each a (a1, a2, , an) in the product, the
  n-tuple a-1 ((a1)-1, (a2)-1, , (an)-1) is the
  inverse of a.

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Direct Sums of Groups

• Sometimes we call the direct product a direct
  sum, especially if we use additive notation.
• A direct product is characterized by its
  projection maps. These turn out to be
  homomorphisms.
• On the other hand, the direct sum is
  characterized by injection maps ji Gi ? G1G2
  Gn that each take ai in Gi to the n-tuple (e1,
  e2, , ai, , en) that has identities in all
  components except for the ith, which has ai.
  These also turn out to be homomorphisms.

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Examples

• Z2Z3
• Z3Z5
• Z2Z2
• Z3Z3
• Z2Z6
• Z9Z6

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Internal Products

• Each component group of a direct product can also
  be considered a subgroup by injection. All of
  these subgroups commute with each other. Thus
  any element g of the product G1G2 Gn can be
  uniquely written in the form g g1 g2 gn with
  gi in Gi. When this happens, G is said to be an
  internal product of the subgroups Gi.

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LCM and GCD of two numbers

• Let x and y be integers, then
• LCM(x, y) is the least multiple of x and y
• LCM(x, y) min m in Z m is a multiple of x
  and m is a multiple of y
• GCD(x, y) is the greatest common divisor of x and
  y
• GCD(x, y) max m in Z m divides x and m
  divides y
• LCM(x, y) x y /GCD(x, y).

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Methods of finding LCM and GCD

• Euclidean Algorithm to find GCD in the form a x
  b y Start with positive integers x and y. Set
  r0 x, r1 y. Given rk-1 and rk, find qk and
  rk1 such that rk-1 qk rk rk1, with 0
  rk1 lt rk. Continue until rk1 0. Then GCD(x,
  y) rk. Then work backward to write GCD in the
  form a x b y.
• Example Find GCD of 64 and 58.
• For small numbers use prime factorization.