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How many distinguishable regular tetrahedral dice can be made where each face has one of the numbers 1,2,3,4 on it? Here two such dice are deemed to be indistinguishable if one can be obtained from the other by a rotation. What about if we allow arbitrary symmetries between indistinguishable such dice? CHAPTER 3 Arithmetic functions 1. 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.

Before showing how this result allows us to determine some concrete examples, we give a generalization. 15. Let X and Y be sets and suppose that Y has a subset Z ⊆ Y which admits a bijection g : Z −→ P(X). Then there is no surjection X −→ Y . Proof. Suppose that f : X −→ Y is a surjection. Choosing any element p ∈ P(X) and defining the function H : X −→ P(X); h(x) = g(f (x)) if f (x) ∈ Z, p if f (x) ∈ / Z, we easily see that h is a surjection, contradicting Russell’s Paradox. Thus no such surjection can exist.

We need the following facts. i) For x, y ∈ G, xH ∩ yH = ∅ ⇐⇒ xH = yH. This is seen as follows. If xH = yH then xH ∩ yH = ∅. Conversely, suppose that xH ∩ yH = ∅. If yh ∈ xH for some h ∈ H, then x−1 yh ∈ H. For k ∈ H, x−1 yk = (x−1 yh)(h−1 k), which is in H since x−1 yh, h−1 k ∈ H and H is a subgroup of G. Hence yH ⊆ xH. Repeating this argument with x and y interchanged we also see that xH ⊆ yH. Combining these inclusions we obtain xH = yH. ii) For each g ∈ G, |gH| = |H|. If gh = gk for h, k ∈ H then g −1 (gh) = g −1 (gk) and so h = k.