Cyclic Groups and Group Homomorphisms

GRE Math Subject · Axiom Academy

LESSON Cyclic Groups and Group Homomorphisms Generators, orders, and structure-preserving maps between groups Definition and Structure of Cyclic Groups A group is cyclic if there exists an element such that every element of is a power of . The two types of cyclic groups: Infinite cyclic: . Every infinite cyclic group is isomorphic to . Finite cyclic of order : . Every cyclic group of order is isomorphic to . Divisors of 12: -- so exactly 6 subgroups Monomorphism = injective homomorphism Epimorphism = surjective homomorphism Isomorphism = bijective homomorphism Automorphism = isomorphism from to itself The Most Important Theorem in Algebra GRE-Style Problem: Let be defined by . Find and .

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