Math 3121 Abstract Algebra I

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Math 3121Abstract Algebra I

• Lecture 14
• Sections 15-16

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Section 15 Factor Groups

• Examples of factor groups
• ZnZm/lt(0,1)gt
• G1G2/i1 (G1) and G1G2/i2 (G2)
• Z4Z6/lt(2,3)gt
• Th Factor group of a cyclic group is cyclic
• Th Factor group of a finitely generated group is
  finitely generated.
• Def Simple groups
• Alternating group An, for 5 n, is simple
  (exercise 39)
• Preservation of normality via homomorphisms
• Def Maximal normal subgroup
• Th M is a maximal normal subgroup of G iff G/M
  is simple
• Def Center
• Def Commutator subgroup

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ZnZm/lt(0,1)gt

• This is isomorphic to Zn
• Note that lt(0,1)gt injects Zm into ZnZm

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Factoring by factors

• Theorem G1G2/i2 (G2) is isomorphic to G1
• Proof Let H i2 (G2) (e, y) y in G2.
  Then (x, e)H (x, y) y in G2. Let p1(x, y)
  x. This is a homomorphism with kernel H and
  image G1. By the Fundamental Theorem of
  Homomorphisms, G1/H is isomorphic to G1.
• Theorem G1G2/i1 (G1) is isomorphic to G2
• More generally?

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Z4Z6/lt(2, 3)gt

• In class
• Order of lt(2, 3)gt
• Order of Z4Z6/lt(2, 3)gt

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A Factor group of a cyclic group is cyclic

• Theorem A Factor group of a cyclic group is
  cyclic
• Proof The image of a generator generates the
  image.

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A Factor group of a finitely generated group is
finitely generated.

• Theorem A Factor group of a finitely generated
  group is finitely generated.
• Proof The image of a generator set generates the
  image.

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Simple groups

• Definition A group is simple if it is nontrivial
  and has no nontrivial normal subgroups.

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Alternating group An, for 5 n, is simple

• Theorem The alternating group An, for 5 n, is
  simple.
• Proof exercise 39

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Preservation of normality via homomorphisms

• Theorem Let h G ? G be a group homomorphism.
  If N is a normal subgroup of G then hH is
  normal in hG. If N is a normal subgroup of
  hG, then h-1N is a normal subgroup of G.
• Proof exercises 35 and 36

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Maximal normal subgroup

• Definition A Maximal normal subgroup M of a
  group G is a normal subgroup is a proper normal
  subgroup such that no proper normal subgroup of G
  contains M.

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The Factor group by a maximal normal subgroup is
simple

• Theorem M is a maximal normal subgroup of G iff
  G/M is simple
• Proof Use the previous theorem

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Center of a Group

• Definition The center of a group G is the set
  c in G c g g c, for all g in G.

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Center

• Theorem The center of a group is an abelian
  subgroup.
• Proof Exercise 52, section 5

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Commutator subgroup

• Definition The commutator subgroup of a group is
  the subgroup generated by all elements of the
  form a b a-1 b-1.

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Commutator subgroup

• Theorem The commutator subgroup C of a group G
  is a normal subgroup of G. If N is a normal
  subgroup of G, then G/N is abelian iff N ? C.
• Proof in book

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HW

• Hand in Nov 25
• Pages 151 4, 6, 8, 14, 35, 36
• Dont hand in
• Pages 151- 1, 3, 5, 7, 9, 13, 15, 39

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Section 16 Group Actions

• Notion of Group Action
• Isotropy Subgroups
• Orbits under a group action

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Group Action

• Definition Let X be a set and G be a group. An
  action of G on X is a map G X ?X such that
  (using infix notation with juxtaposition)
• 1) e x x for all x in X
• 2) (g1 g2)(x) g1 (g2 x) for all x in X and g1
  g2 in G.
• Notation In the above we write (g, x) g x.
• Definition A G-set is a set X together with an
  action of G on X.

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Examples

• Let X be any set, and let H be any subgroup of
  permutations on X. Define an action
• G X ? X
• by
• (p, x) p(x)
• or
• p x p(x)
• Then
• 1) e x e(x) x
• 2) (p1 p2)(x) p1(p2(x)) (composition)

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Actions are Permutations

• Theorem Let X be a G-set. For each g in G, the
  function sg X ? X defined by
• sg(x) g x, for x in X
• is a permutation of X.
• Also the map s G ? SX defined by
• f(g) sg, for g in G
• is a homomorphism with the property that
• f(g)(x) sg(x) g x
• Proof in the book

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Faithful and Transitive Actions

• Definition Let X be a G-set. If e is the only
  member that fixes all x in X, then G acts
  faithfully on X.
• Definition A group is transitive on a G-set X,
  if for each x1, x2 in X, there is a g in G such
  that
• g x1 x2.

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More Examples

• Every group G is itself is a G set with the
  action given by the binary group operation.
• Left cosets of a subgroup.
• Dihedral groups (look at D4)

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Isotropy Group

• Notation Let X be a G-set and define
• Xg x in X g x x
• Gx g in G g x x
• Theorem Let X be a G-set. Then Gx is a group
  for all x in X.
• Definition Gx is called the isotropy group of x.

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Orbits

• Theorem Let X be a G-set. Define a relation on
  X by
• x1 x2 ? g x1 x2 for some g in g
• Then is an equivalence relation on X.
• Proof (Outline)
• 1) reflexive because e is in G
• 2) symmetric because G is closed under inverses.
• 3) transitive because G is closed under
  multiplication.

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Orbits

• The equivalence classes of this equivalence
  relation are called orbits under the action.

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Lagrange Revisited

• Theorem Let X be a G-set and let x be in X. The
  G x (G Gx ). If G is finite, then G
  x divides G.
• Proof Define a map h from G x onto G/Gx, the
  collection of left cosets of Gx in G by
• h(y) g Gx ? y g x
• This is well-defined, 1-1, and onto.
• Well-defined
• y in G x ? y g x for some g in G
• Suppose y g1 x and y g2 x.
• Then g1 x g2 x ? g1-1g1 x g1-1g2 x
• ? e x g1-1g2 x ? x g1-1g2 x ? g1-1g2 in
  Gx
• Thus g1 Gx g2 Gx.
• 1-1 Suppose h (y1) h (y2), for y1 and y2 in
  G x. Then there are g1 and g2 such that
• y1 g1 x and y2 g2 x. Since h (y1) h
  (y2), g1 Gx g2 Gx. And so on.(see book)
• onto (see book)

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HW Section 16

• Dont hand in
• Page 159- 1, 2, 3