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If σ decomposes into an even number of transpositions (that is, sign(σ ) = +1) we say it is an even permutation. Otherwise we call it an odd permutation. 31 32 a course in abstract algebra The product of two odd permutations or two even permutations is clearly going to be an even permutation, while the product of an odd permutation with an even one is clearly going to be odd. Furthermore, the parity (odd or even) of an inverse permutation σ−1 will be equal to the parity of the original permutation σ.

This has order–n rotational symmetry, and n axes of symmetry. In the case where n is odd, each of these axes passes through the midpoint of one side and the opposite vertex. 5: The multiplication table for the dihedral group D3 through pairs of opposite vertices, and the other half pass through the m2 midpoints of opposite sides. 43 The dihedral group Dn = {e, r, . . , r n−1 , m1 , . . , mn } is the symmetry group of the regular plane n–gon. Here, e is the identity, r represents an anticlockwise rotation through an angle of 2π n , and mk denotes a reflection in the line which makes an angle of kπ n with the horizontal.

How many permutations in S7 have this order? For n ∈ Z define a function f n : Z → Z by f n (m) = m + n for any m ∈ Z. Show that f n ∈ Sym(Z), and also that f n ◦ f m = f m+n = f m ◦ f n and f n−1 = f −n for any m, n ∈ Z. (a) Show that the function f : Z13 → Z13 given by f ( x ) = x5 is a permutation. Write this permutation in disjoint cycle notation. (b) More generally, choose some integer 0 < α < 13, show that the Dickson polynomial D5 ( x, α) = x5 − 5αx3 + 5α2 x defines a permutation of Z13 , and write this permutation in disjoint cycle notation.

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A course in abstract algebra by Nicholas Jackson


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