Let ne N. Show that Zn forn > 2 is not a group under multiplication as...
Exercise 1.25. This exercise relates to (1.13). Suppose that x > 1. For each ne N let y= Vå be defined by yn = x. This implies that vx < mă if n > m. To prove that Veso 3NEN Vnən : 0< V2-1<e it therefore suffices to prove that Vesonen: 0< Vx-1<e. Prove this latter statement.
Let ne Nj. Prove that n < 2(6(n)).
b. Let U2 u~xã. Show that E(b)=n-2 for n > 2. LU
n+1 2. Let an = (-1)"-for n > 1. (a) What is sup(fas, ab,...])? (b) What is supa,)? (d) What is liminfan-sup ((inf((an.an+1, )) | n = 1,2.. ..))?
Q 3 a) Let n > 2 be an integer. Prove that the set {z ET:z” = 1} is a subgroup of (T, *). Show that it is isomorphic to (Zn, + mod n). b) Show that Z2 x Z2 is not isomorphic to Z4. c) Show that Z2 x Z3 is isomorphic to 26.
IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .
5. Prove that U(2") (n > 3) is not cyclic.
ei0 : 0 E R} be the group of all complex numbers on the unit circle under multiplication. Let ø : R -> U 1. (30) Let R be the group of real numbers under addition, and let U be the map given by e2Tir (r) (i) Prove that d is a homomorphism of groups (ii) Find the kernel of ø. (Don't just write down the definition. You need to describe explicit subset of R.) an real number r for...
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
3. Let {Sn, n > 0} be a symmetric Random Walk on Z. Defined To inf(n > 1 : Sn-0} the time of first passage to state 0, prove that 2n - 1 for any n 2 1