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Unformatted text preview: 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 ∈ S n have the same cycle structure IFF there exists α ∈ S n such that γ = 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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