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Notice that {+1, −1} is actually a group under multiplication. 7. The function sgn : Sn −→ {+1, −1} satisfies sgn(τ σ) = sgn(τ ) sgn(σ) (τ, σ ∈ Sn ). Proof. By considering the arrow diagram for τ σ obtained by joining the diagrams for σ and τ , we see that the total number of crossings is cσ + cτ . If we straighten out the paths starting at each number in the top row, so that we change the total number of crossings by 2 each time. So (−1)cσ +cτ = (−1)cτ σ . A permutation σ is called even if sgn σ = 1, otherwise it is odd.

Gp1 is usually called the associativity law. ι is usually called the identity element of (G, ∗). In Gp3, the unique element y associated to x is called the inverse of x and is denoted x−1 . 1. The following are examples of groups. (1) G = Z, ∗ = +, ι = 0 and x−1 = −x. (2) G = Q, ∗ = +, ι = 0 and x−1 = −x. (3) G = R, ∗ = +, ι = 0 and x−1 = −x. 2. Let n > 0 be a natural number. Then (Z/n, +) is a group with ι = 0n , x −1 = −xn = (−x)n . 3. Let R = Q, R, C. Then each of these choices gives a group (GL2 (R), ∗) with GL2 (R) = a b : a, b, c, d ∈ R, ad − bc = 0 , c d ∗ = multiplication of matrices, 1 0 = I2 , 0 1   d b −  ad − bc  .

Consider the equation td θ(k/d) = δ(k) = 0. d|k Rewriting this as tk = − td θ(k/d), d|k d=k ˜ we see that tk is uniquely determined from this equation. Now define θ˜ by θ(n) = tn . By construction, ˜ ˜ = δ(n). θ ∗ θ(n) = θ(n/d)θ(d) d|n ˜ By (d) we also have θ˜ ∗ θ = θ ∗ θ. (d) We have θ ∗ ψ(n) = θ(d)ψ(n/d) = d|n ψ(k)θ(n/k) = ψ ∗ θ(n). ψ(n/d)θ(d) = d|n k|n ˜ In each of the groups (AFR , ∗) and (AFC , ∗), the inverse of an arithmetic function θ is θ. Here is an important example. 5. The inverse of η is η˜ = µ, the M¨ obius function.

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