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Exam2Review - Exam 2 Review 1 Preliminaries(a Division...

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Exam 2 Review 1. Preliminaries (a) Division Algorithm (b) gcd, lcm (c) basics of modular arithmetic. 2. Symmetric Groups (i.e. Permutation Groups) (a) order of a permutation (b) inverse of a permutation (c) even/odd permutations (d) permutations as products of disjoint cycles (e) | S n | = n ! (f) γ, γ 0 S n have the same cycle structure IFF there exists α S n such that γ 0 = αγα - 1 . 3. Groups (a) Requirements for ( X, * ) to be a group: i. Closure ii. Associativity iii. Identity iv. Inverses (b) If ( X, * ) is NOT a group, you should be able to quickly recognize which condition fails. (c) Cayley Tables (d) Elementary Properties of Groups (e) ( ab ) - 1 = b - 1 a - 1 (f) Order of an element, Order of a group (g) If x G , o ( x ) = k and x n = e then k | n . (h) Dihedral Groups 4. Subgroups and Cyclic Grops (a) Definition of a subgroup (b) One-step subgroups test (really 2-steps) i. H 6 = Ø ii. a, b H ab - 1 H . (c) Definition of cyclic group/subgroup (d) If o ( a ) = n then | h a i | = n . (e) Let G = h a i be a cyclic group of order n . Then G = a k ( k, n ) = 1 (f) Fundamental Theorem of Cyclic Groups i. Every subgroup S of a cyclic group G = h a i is itself cyclic.
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