Download Math 3121 Lecture 14

Math 3121
Abstract Algebra I
Lecture 14
Sections 15-16
Section 15: Factor Groups
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Examples of factor groups
ℤn×ℤm/<(0,1)>
G1×G2/i1 (G1) and G1×G2/i2 (G2)
ℤ4×ℤ6/<(2,3)>
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
ℤn×ℤm/<(0,1)>
• This is isomorphic to ℤn
• Note that <(0,1)> injects ℤm into ℤn×ℤm
Factoring by factors
Theorem: G1×G2/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: G1×G2/i1 (G1) is isomorphic to G2
More generally?
ℤ4×ℤ6/<(2, 3)>
• In class
• Order of <(2, 3)>
• Order of ℤ4×ℤ6/<(2, 3)>
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.
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.
Simple groups
• Definition: A group is simple if it is nontrivial
and has no nontrivial normal subgroups.
Alternating group An, for 5≤ n, is simple
• Theorem: The alternating group An, for 5≤ n,
is simple.
• Proof: exercise 39
Preservation of normality via homomorphisms
Theorem: Let h: G  G’ be a group
homomorphism. If N is a normal subgroup of
G then h[H] is normal in h[G]. If N’ is a normal
subgroup of h[G], then h-1[N’] is a normal
subgroup of G.
Proof: exercises 35 and 36
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.
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
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.
Center
Theorem: The center of a group is an abelian
subgroup.
Proof: Exercise 52, section 5
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.
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
HW
• Hand in Nov 25:
Pages 151: 4, 6, 8, 14, 35, 36
• Don’t hand in:
Pages 151-: 1, 3, 5, 7, 9, 13, 15, 39
Section 16: Group Actions
• Notion of Group Action
• Isotropy Subgroups
• Orbits under a group action
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.
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)
Actions are Permutations
Theorem: Let X be a G-set. For each g in G, the
function: σg : X  X defined by
σg(x) = g x, for x in X
is a permutation of X.
Also the map σ: G  SX defined by
φ(g) = σg, for g in G
is a homomorphism with the property that
φ(g)(x) = σg(x) = g x
Proof: in the book
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.
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)
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.
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.
Orbits
• The equivalence classes of this equivalence
relation are called orbits under the action.
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=gx
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)
HW Section 16
• Don’t hand in
Page 159-: 1, 2, 3