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MA342A (Harmonic Analysis 1) Tutorial sheet 2 [October 22, 2015] Name: Solutions 1. For G = Z4 the cyclic group of order 4, make a table of all the values of (all) the characters χ ∈ Ĝ. Solution: Write G = {0, 1, 2, 3} with addition modulo 4. We know that the characters are given by χ(1) which must be a 4th root of unity. That is χk (1) = e2πik/4 for k = 0, 1, 2, 3. e2πi/4 = i and so we get the values for χk (n) = (χk (1))n (0 ≤ k, n < 4) n χ0 (n) χ1 (n) χ2 (n) χ3 (n) 0 1 2 3 1 1 1 1 1 i −1 −i 1 −1 1 −1 1 −i −1 i P Then compute the sum χ∈Ĝ χ P P Solution: χ∈Ĝ χ(n) = 4 if n = 0 and χ∈Ĝ χ(n) = 0 for n = 1, 2, 3. 2. For G = Z with the usual absolute value metric d(m, n) = |m − n|, let ρ be the metric on G×G with ρ((m1 , m2 ), (n1 , n2 )) = d(m1 , n1 )+d(m2 , n2 ). Show that the metric topology on (G × G, ρ) is the discrete topology. Solution: Since d(m, n) ≥ 1 if m 6= n, it follows that ρ((m1 , m2 ), (n1 , n2 )) ≥ 1 if ((m1 , m2 ) 6= (n1 , n2 ) (since then either m1 6= n1 or m2 6= m2 or both). Thus B((m, n), 1) = {(m, n)} and every point of every subset of Z × Z is an interior point. So every subset of Z × Z is open in the ρ topology. So the ρ topology is the discrete topology. ∼ 3. Show that the result G\ 1 ⊕ G2 = Ĝ1 × Ĝ2 for finite abelian groups G1 and G2 also holds for the case G1 = G2 = Z. Solution: The reasoning in the notes: If χ : G1 ⊕ G2 → T is a character, then χ1 (g1 ) = χ(g1 , 0) gives a character of G1 and χ2 (g2 ) = χ(0, g2 ) gives a character of G2 . In fact we can recover χ from χ1 and χ2 because χ(g1 , g2 ) = χ((g1 , 0) + (0, g2 )) = χ(g1 , 0)χ(0, g2 ) = χ1 (g1 )χ2 (g2 ) From this we can also see that if χ1 ∈ Ĝ1 and χ2 ∈ Ĝ2 , then the above formula gives χ ∈ G\ 1 ⊕ G2 and also that χ is uniquely determined by (χ1 , χ2 ). It follows that we can identify ∼ G\ 1 ⊕ G2 = Ĝ1 × Ĝ2 and we can also see that the group structure on G\ 1 ⊕ G2 corresponds to the direct product group operation on Ĝ1 × Ĝ2 , the one where (χ1 , χ2 ) · (χ01 , χ02 ) = (χ1 χ01 , χ2 χ02 ) This is actually the same as the direct sum of Ĝ1 and Ĝ2 but the direct sum notation is best used for groups written additively. was for finite groups, but in fact works more generally. In the case of toplogical groups, we need to be concerned that characters are not just homomorphisms but also are required to be continuous. Since Z and Z × Z are discrete, continuity imposes no additional condition. So the same argument works. Richard M. Timoney 2