Abstract Algebra Exam 1 Review Problems and Solutions

   • Abstract Algebra Course, Lecture 1: Introd...  . Review of basic Group Theory: number theory, equivalence relations, group definition, subgroups, subgroup test, order of a subgroup, order of an element, Abelian groups, cyclic groups, permutation groups, and isomorphisms. https://amzn.to/2ZqLc1J ("Contemporary Abstract Algebra", by Joe Gallian)

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(0:00) Introduction
(0:47) a divides b definition
(2:10) Euclid's Lemma
(4:01) Relatively prime definition
(5:33) Group definition
(10:22) Center of a group definition
(12:56) Isomorphism definition
(16:56) Are cyclic groups Abelian?
(18:46) Are Abelian groups cyclic?
(21:53) Is D3 (dihedral group) cyclic? (D3 is the symmetries of an equilateral triangle)
(26:47) GCD is a linear combination theorem
(27:30) If |a| = 6, is a^(-8) = a^(4)? (the order of "a" is 6)
(29:21) Do the permutations (1 3) and (2 4) commute? (they are disjoint cycles)
(30:21) Is the cycle (1 2 3 4) an even permutation?
(31:56) Number of elements of order 2 in S4, the symmetric group on 4 objects
(36:19) Generators of the cyclic group Z24. Relationship to U(24). Euler phi function value φ(24).
(42:06) If |a| = 60, answer questions about (a) (cyclic subgroup generated by a): possible orders of subgroups, elements of (a^12), order |a^12|, order |a^45|.
(49:17) Permutation calculations, including the order of the product of disjoint cycles as the lcm of their orders (least common multiple of their orders)
(54:51) One-step subgroup test to prove the stabilizer of an element under a permutation group is a subgroup of that permutation group.
(1:02:55) Induction proof that φ(a^n) = (φ(a))^n for all positive integers n.
(1:08:25) Direct image of a subgroup is a subgroup (one-step subgroup test).
(1:13:52) Prove a relation is an equivalence relation. Find equivalence classes. (Related to modular arithmetic).

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