Chapters 5 and 6

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Chapters 5 and 6

Chap 5 S_n is non-Abelian for all n >= 3.

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Chap 5 If a is a permutation that is an m-cycle and b is a permutation that is an n-cycle, then |ab| = lcm(m,n).

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Chap 5 A permutation that is a product of m even permutations and n odd permutations is an even permutation if and only if n is even.

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Chap 6 Every group is isomorphic to a group of permutations.

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Chap 6 A 1-1 mapping takes distinct elements to distinct elements.

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Chap 6 A 1-1 mapping from a set to itself is onto.

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Chap 6 Every infinite cyclic group is isomorphic to Z.

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Chap 6 Every two cyclic groups of the same order are isomorphic.

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Chap 6 D_n has a subgroup isomorphic to Z_n.

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Chap 6 A group can be isomorphic to a proper subgroup of itself.

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Chap 6 Two groups isomorphic to the same group are isomorphic to each other.

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