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# ABSTRACT ALGEBRA I MATH 301

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This 3 page Class Notes was uploaded by Ms. Helen Sipes on Sunday September 27, 2015. The Class Notes belongs to MATH 301 at Iowa State University taught by Staff in Fall. Since its upload, it has received 5 views. For similar materials see /class/214507/math-301-iowa-state-university in Mathematics (M) at Iowa State University.

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Date Created: 09/27/15

Review topics for Math 301 April 22 1999 1 De nitions 0 Group actions transitive faithful orbit stabilizer left regular represen tation Conjugation conjugacy conjugate class centralizer center 0 Automorphism inner automorphism normalizer of a subgroup o Dihedral group frieze groups Euclidean group En orthogonal groups 001 50m 0 Parity alternating group An conjugacy in Sn 2 Some Theorems 0 The order of the orbit is the index of the stabilizer o Burnside s Theorem 0 Sylowls Theorem extended version 0 Structure of E2 and E3 0 Odd and even permutations 3 Exercises 1 Equivalence relations have appeared many times in this course Name three examples of equivalence relations In each case tell What the equivalence classes are and state a theorem that applies in that particular case the general theo rem that equivalence classes partition the set on Which the relation is de ned does not count 2 Show that the map I gt gt exp27riz is a homomorphism from R into C and use it to deduce that the circle group the group of complex numbers of absolute value 1 is isomorphic to RZi 3 3414 a b 34 1635i1137i8i MATHEMATICS 301 Section B 0 Chapter 6 1 Understand de nition and properties of isomorphism and use them to prove theorems 2 Understand Caley s theorem 3 Understanding automorphisms and inner automorphisms of groups 4 Be able to calculate AutG or InnG 0 Chapter 7 1 Understand left and right cosets and be able to calculate explicit cosets 2 Know properties of cosets and use them to prove theorems 3 Understand Langrange s theorem and its consequences and use them to prove theorems 4 Know the classi cation of groups of order 2p 5 Stabilizer of a point and orbit of a point 0 Chapter 8 7 De nition and properties of external direct product 7 Know how to calculate the order of an element in a direct product 7 Know how to tell whether a direct product is cyclic 7 Know Theorem 83 and the proof 0 Chapter 9 1 Know the de nition of normal subgroup 2 Know how to use theorem 91 to verify whether a given subgroup is normal or not 3 Understand the factor group and know how to write down the elements in a factor group Know how to multiply two elements in a factor group 4 Know applications of factor group including GZG z InnG and Cauchy s theorem for abelian group 0 Chapter 9 1 De nition of group homomorphism7 kernel of a homomorphism 2 Know the properties of homomorphisms and be able to use them to prove theorems 3 Know Kernels are normal 4 Know the rst isomorphism theorem 0 Chapter 11 1 Know the fundamental theorem of nite abelian groups and use it to prove theorems 2 Given natural number n be able to list all non isomorphic structures for abelian groups with order n Review topics for Math 301 1 De nitions 0 Mapping composition one to one onto permutation cycle 0 Group operation associativity identity inverse subgroup commutative abelian group cyclic group 0 Subgroup invariant set orbit 0 Relation equivalence re exive symmetric transitive partition equiva lence c ass 0 Congruence congruence class greatest common divisor relatively prime 2 Some Groups 0 Dihedral groups Dn o Symmetric groups Sn 0 Additive number groups Z77 Zn o Multiplicative number groups Q 0 R 0 23f 0 General linear group Q 2 R 02 3 Exercises in proof 1 Let X be a nite set a z X A X a mapping Then a is one to one if and only if a is onto 2 Let A be a 2 X 2 invertible real matrix The mapping of R2 de ned by 11 AI is a permutation of R2 Thus Q 2R and 02 can be viewed as groups of permutations of R i 3 Let n E Z The set of all integer multiples of n is a subgroup of Z

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