Given: G is the group of 31st roots of unity under complex multiplication. Z31 is the group of integers mod 31 under modular addition. Let elements in 31st roots of unity be represented with Euler's formula: cos(2x / 31) + i sin(2x / 31) = e^2ix/31 The function defined by f (e^2ix/31 ) = [x]31 is a homomorphism from G -> Z31 A. Prove that the given function is operation preserving. SOLUTION 2 x 2 x cos + isin =e 31 31 ( ) ( ) let : 2 ix 31 2 x a2 ix = =b 31 31 form maclaurin series ; 1n a3 a5 a2 n+1=a + ... 3! 5! n=0 ( 2 n+1 ) ! sin a= 1n 2 n a2 a 4 a =1 + ... 2! 4 ! n=0 2 n! cos a= e b= n=0 bn z2 z3 =1+ z + + + ... n! 2! 3 ! therefore 2 e 2 ix 31 = n=0 3 2 ix 2 ix 2 ix 31 31 31 2 ix =1+ + + +... n! 31 2! 3! ( ) ( ) 2 3 4 5 2 x 2 ix 2 ix 2 ix 31 31 31 31 2 ix 1+ i + +i ... 31 2! 3! 4! 5! ( ) ( ) ( ) ( ) 2 x 2 2 ix 4 2 ix 3 2 ix 5 31 31 31 31 2 ix 1 + ... +i + ... 2! 4! 31 3! 5! ( cos ( ) ( ) )( ( ) ( ) ) ( 231x )+isin ( 231x ) Therefore since the operation + is maintained on both sides of the equation, then the function is operation preserving