# Get Algebra: A Computational Introduction PDF

By John Scherk

ISBN-10: 1584880643

ISBN-13: 9781584880646

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Ak a1 . . ak a1 . . Prove that n | (10k − 1). 9. For each prime p < 30, find the smallest k ∈ N such that 10k + 1 ≡ 0 (mod p) , and compare it with the length of the period of 1/p. 10. Let p be a prime number, and let k be the length of the period of 1/p. Suppose that k = 2l is even. Prove that 10l + 1 is divisible by p, and that 10m + 1 ( is not divisible by p for any m < l. Suggestion: write 102l − 1 = (10l − ) 1)(10l + 1). Does this hold true if p is a composite number? 11. • Make a table of φ(n) for n ≤ 20.

So the set D4 will have the same properties. You can also check this directly. Sets of permutations with these algebraic properties are called permutation groups. As we shall see they arise in many contexts. 1. A non-empty set of permutations G ⊂ Sn is called a permutation group (of degree n) if for all α, β ∈ G (i) αβ ∈ G (ii) α−1 ∈ G . Sn itself is a permutation group, called the full permutation group (of degree n) or symmetric group (of degree n). Another example is V ′ = {(1 2), (3 4), (1 2)(3 4), (1)} ⊂ S4 .

8. 4. EXERCISES a) ( ) 1 2 3 4 5 6 7 8 9 4 6 7 1 5 2 8 3 9 b) ( ) 1 2 3 4 5 6 7 8 9 6 1 7 5 4 2 8 9 3 9. Write the two permutations in the previous exercise as products of transpositions. 10. Show that the inverse of an even permutation is even, of an odd permutation odd. 1 Definition Suppose you have a square and number its vertices. 1 . 4 Each symmetry of the square permutes the vertices, and thus give you an element of S4 . We can make a table showing the 8 symmetries of the square and the corresponding permutations: Symmetry Permutation rotation counterclockwise through π/2 rotation counterclockwise through π rotation counterclockwise through 3π/2 identity map reflection in diagonal through 1 and 3 reflection in diagonal through 2 and 4 reflection in vertical axis reflection in horizontal axis (1 2 3 4) (1 3)(2 4) (1 4 3 2) (1) (2 4) (1 3) (1 2)(3 4) (1 4)(2 3) 35 36 CHAPTER 3.

### Algebra: A Computational Introduction by John Scherk

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