Math417Midterm2_ReviewSheet - )Theorem 6.4 Aut G and Inn G...

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MATH 417 Midterm 2 Study Guide The main focus of the exam is on chapters 5, 6, 7, 8 and Chapter 10 pages 199-201. Definitions you should know: permutation symmetric group of degree n cycle, k-cycle transposition even/odd permutations Alternating group of degree n kernel, image, inverse image isomorphism homomorphism automorphism identity homomorphism trivial homomorphism inner automorphism Aut ( G ) Inn ( G ) left (right) coset of H in G containing a index of H in G external direct product Theorems you should know the statements of and be able to use: Theorem 5.1 Products of disjoint cycles Theorem 5.2 Disjoint cycles commute Theorem 5.3 Order of a Permutation (*)Theorem 5.4 Product of 2-cycles Theorem 5.5 Always Even or Always Odd (*)Theorem 5.6 Even permutations form a group (*)Theorem 5.7 Theorem 6.1 Cayley’s Theorem (*)Theorem 6.2 and also Theorem 10.1 parts 1)-4) (see notes from class) (*)Theorem 6.3 and also Theorem 10.2 parts 1)-3) (see notes from class)
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Unformatted text preview: (*)Theorem 6.4 Aut ( G ) and Inn ( G ) are Groups (*)Lemma pg. 138 Properties of Cosets Theorem 7.1 Lagrange’s Theorem Corollary 1 [ G : H ] = | G | | H | (*)Corollary 2 | a | Divides | G | (*)Corollary 3 Groups of Prime Order are Cyclic (*)Corollary 4 a | G | = e Theorem 8.1 Order of an Element in a Direct Product Theorem 8.2 Criterion for G ⊕ H to Be Cyclic Corollary 1 pg. 156 Criterion for G 1 ⊕ G 2 ⊕ ··· ⊕ G n to Be Cyclic Corollary 2 pg. 156 Criterion for Z n 1 n 2 ··· n k = Z n 1 ⊕ Z n 2 ⊕ ··· ⊕ Z n k • Here are some practice problems from the textbook: Chapter 5: 2, 3, 4, 6, 7, 9, 12, 15, 17, 18, 22, 30, 38, 51 Chapter 6: 1, 3, 5, 11, 15, 19, 22, 25, 26, 35 Chapter 7: 1, 2, 3, 5, 7, 8, 13, 14, 23, 25, 26, 38, 39 Chapter 8: 2, 3, 5, 7, 8, 9, 11, 14, 17, 21, 23, 25, 31, 41 Chapter 10: 1, 2, 3, 4, 7...
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