Math 3121 Abstract Algebra I Lecture 14 Sections 15-16

Section 15: Factor Groups Examples of factor groups ℤ n × ℤ m / G 1 ×G 2 /i 1 (G 1 ) and G 1 ×G 2 /i 2 (G 2 ) ℤ 4 × ℤ 6 / 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 A n, 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

ℤ n × ℤ m / This is isomorphic to ℤ n Note that injects ℤ m into ℤ n × ℤ m

Factoring by factors Theorem: G 1 ×G 2 /i 2 (G 2 ) is isomorphic to G 1 Proof: Let H = i 2 (G 2 ) = {(e, y) | y in G 2 }. Then (x, e)H = {(x, y) | y in G 2 }. Let p 1 (x, y) = x. This is a homomorphism with kernel H and image G 1. By the Fundamental Theorem of Homomorphisms, G 1 /H is isomorphic to G 1. Theorem: G 1 ×G 2 /i 1 (G 1 ) is isomorphic to G 2 More generally?

ℤ 4 × ℤ 6 / In class Order of Order of ℤ 4 × ℤ 6 /

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 A n, for 5 ≤ n, is simple Theorem: The alternating group A n, 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) (g 1 g 2 )(x) = g 1 (g 2 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) (p 1 p 2 )(x) = p 1 (p 2 (x)) (composition)

Actions are Permutations Theorem: Let X be a G-set. For each g in G, the function: σ g : X  X defined by σ g (x) = g x, for x in X is a permutation of X. Also the map σ: G  S X defined by φ (g) = σ g, for g in G is a homomorphism with the property that φ (g)(x) = σ g (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 x 1, x 2 in X, there is a g in G such that g x 1 = x 2.

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 D 4 )

Isotropy Group Notation: Let X be a G-set and define: X g = {x in X | g x = x} G x = {g in G | g x = x} Theorem: Let X be a G-set. Then G x is a group for all x in X. Definition: G x is called the isotropy group of x.

Orbits Theorem: Let X be a G-set. Define a relation on X by x 1 ~ x 2 ⇔ g x 1 = x 2 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: G x ). If |G| is finite, then |G x| divides |G|. Proof: Define a map h from G x onto G/G x, the collection of left cosets of G x in G by h(y) = g G x ⇔ 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 = g 1 x and y = g 2 x. Then g 1 x = g 2 x ⇒ g 1 -1 g 1 x = g 1 -1 g 2 x ⇒ e x = g 1 -1 g 2 x ⇒ x = g 1 -1 g 2 x ⇒ g 1 -1 g 2 in G x Thus g 1 G x = g 2 G x. 1-1: Suppose h (y 1 ) = h (y 2 ), for y 1 and y 2 in G x. Then there are g 1 and g 2 such that y 1 = g 1 x and y 2 = g 2 x. Since h (y 1 ) = h (y 2 ), g 1 G x = g 2 G x. And so on.(see book) onto: (see book)

HW Section 16 Don’t hand in Page 159-: 1, 2, 3