Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Birkhoff's representation theorem wikipedia , lookup
Algebraic K-theory wikipedia , lookup
Laws of Form wikipedia , lookup
Oscillator representation wikipedia , lookup
Homomorphism wikipedia , lookup
Congruence lattice problem wikipedia , lookup
Coxeter notation wikipedia , lookup
Group action wikipedia , lookup
Math 3121 Abstract Algebra I Lecture 14 Sections 15-16 Section 15: Factor Groups • • • • • • • • • • • • • 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