Question

For modern/abstract algebra.

[2] For each divisor k ofn, let U.(n){x EU( D List the elements of U,(20) i) Show that U. (n)is a subgroup of U(n) mod k-1}

Answer 1

The given information for each division k of n u_k(n) = {x elementof u(n): x (mod k) = 1} (i) u_4(20) = {x elementof U(2): x (mod 4) = 1} u(2) = {1, 3, 7, 9, 11, 13, 17, 19} u_4(20) = {9, 13, 17} [divide elements of u(2) by 4 & remainder should be 1} (ii) show that u(n) is a sub group of u(n) Let g elementof u(n) , h elementof u_k(n) h elementof u(n) & h (mod k) = 1 To show ghg^-1 elementof u_k (n) ghg^-1 (mod k) = 1 (ghg^-1 mod 19)(mod k) = 1 Let n = k t and ghg^-1 = q_n + r , 0 lessthanorequalto r < n Then (ghg^-1 mod n) (mod k) = (r mod n) (mod k) (because n mod n = 0) = ((ghg^-1` - q_n) (mod n) (mod k) = ghg^-1 - q_n) mod k \r\n = (ghg^-1 - q_kt) mod k = ghg^-1