Homework 5

5

Homework

P. 5.1. Let G be a group and ρ an equivalence relation on G, such that for any x, y, z ∈ G the following implication holds: x ρ y =⇒ x · z ρ y · z . Show that the equivalence class [1]ρ is a subgroup H of G, and that G/ρ = (G/H)r . P. 5.2. A function f : N∗ −→ Z is called multiplicative if for any m, n ∈ N∗ such that (m, n) = 1 the equality f (m · n) = f (m) · f (n) holds. We shall denote by M the set of all multiplicative functions. On M we define the convolution product ∗ by n X . (f ∗ g)(n) = f (d) · g d d|n

a) Show that (M, ∗) is a monoid, with the unit element u : N∗ −→ Z defined by  1 , if n = 1 ; u(n) = 0 , if n > 1 . b) If c : N∗ −→ Z : n 7−→ 1 is the constant 1 function, show that c ∈ U (M) and determine a formula for its inverse c−1 . c) Use the identity X n= ϕ(d) d|n

(where ϕ is Euler’s function) to show that id = ϕ ∗ c. Deduce another formula for ϕ(written in terms of the inverse of c).

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