Good day

. Do you know anyone of this?? pleeease help.. thank you so much!!!

1 2 3 4 5 1. Let S {1,2,3,4,5} and let be a binary 1 1 2 3 4 5 operation on S defined by the table on the right. 2 2 1 4 5 3 (a) Calculate 5 [(304) 2). 3 3 5 1 2 4 (b) Determine all x S such that (3 x) 5 = 2. 4 4 3 5 1 2 (c) Show that (S,) is not a group. 5 5 4 2 3 1 2. Let (G1. *) and (G2..) be groups. Let G X G2 = {(a,b) a E G and b G2}. Define the operation on G1 x G2 as (a, b)(az, b2) = (a + b, a, o b). Show that G1 G2 is a group under . (b) Show that Z4 x Zz is isomorphic to Z 12. (o) Show that Z4 x Z2 is not isomorphic to 28. 3. Let G = (a) be a finite cyclic group of order 90. Determine the following with justification: order of 220 (b) order of a70 (c) order of (220) n (a70) (d) complete set of generators for G (e) qumber of elements x E G such that x77 = a. 4. Consider the group 275. (a) Give a complete list of subgroups for Z75 and sketch the lattice of subgroups. (b) How many generators does Z75 have? (c) Show that the group U(Z75) under multiplication modulo 75 is not cyclic. (d) Determine the complete list of subgroups of U(Z75) (justify why your list is and sketch the lattice of subgroups