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Then we have σ = (k1 σ(k1 ) · · · σ r1 −1 (k1 )) · · · (kd σ(kd ) · · · σ rd −1 (kd )), which is the disjoint cycle decomposition of σ. It is unique apart from the order of the factors and the order in which the numbers within each cycle occur. For example, in S4 , (1 2)(3 4) =(2 1)(4 3) = (3 4)(1 2) = (4 3)(2 1), (1 2 3)(1) =(3 1 2)(1) = (2 3 1)(1) = (1)(1 2 3) = (1)(3 1 2) = (1)(2 3 1). We usually leave out cycles of length 1, so for example (1 2 3)(1) = (1 2 3). Recall that when performing elementary row operations (ERO’s) on n × n matrices, one of the types involves interchanging a pair of rows, say rows r and s, this operation is denoted by 5.

Definition and examples of arithmetic functions Let Z+ = N0 −{0} be the set of positive integers. A function ψ : Z+ −→ R (or ψ : Z+ −→ C) is called a real (or complex) arithmetic function if ψ(1) = 1. There are many important and interesting examples. 1. The following are all real arithmetic functions: a) The ‘identity’ function id : Z+ −→ R; id(n) = n. 24. c) For each positive natural number r, σr : Z+ −→ R; dr . σr (n) = d|n σ1 is often denoted σ; σ(n) is equal to the sum of the (positive) divisors of n.

27. 7. Group actions If X is a set and (G, ∗) then a (group) action of (G, ∗) on X is a rule which assigns to each g ∈ G and x ∈ X and element gx ∈ X so that the following conditions are satisfied. GpAc1 For all g1 , g2 ∈ G and x ∈ X, (g1 ∗ g2 )x = g1 (g2 x). GpAc2 For x ∈ X, ιx = x. Thus each g ∈ G can be viewed as acting as a permutation of X. 29. Let G Sn and let X = n. For σ ∈ G and k ∈ n let σk = σ(k). This defines an action of (G, ◦) on n. 7. 30. Let X ⊆ Rn and let G Sym(X) be a subgroup of the symmetry group of X.