Question

Need help with all parts please. Thank you.

1. Orders. [Purpose: Apply theorems about orders in Zn and in direct products.] (a) Find the order of ([12]28. [8]60,岡63) in the group Z2eX Z60 × Z63. (The notation kn is to emphasize we are working mod n in that particular coordinate.) (b) Find all elements of order 6 in 24 × Z6 or explain why none exist. (c) Find all elements of order 8 in Z Zg or explain why none exist.

Answer 1

(a) We find the order of ([12]_28, [8]_60, [35]_63) in the group Z_28 times Z_60 times Z_63 equation (1) the order of 12 in Z_28 (12)^7 iscongruentto 0 mod (28) 1 mod 28 = exclus_28 hence is the order of 12 in Z_28 is 7 and the order of 8 in 760 (8)^15 iscongruentto 0 mod (60) mod 60 = exclus_63 hence the order of 35 in Z_63 is 9 The order of ([12]_28, [8]_60, [35]_63) = l.c.m (order [12]_28, order [8]_60, order [35]_63) = l.c.m (7, 15, 9) = 315 \r\n (b) We find all elements of order 6 in Z_4 times z_6 z_4_(1, 2, 4) times z_6_(1, 2, 3, 6) possible No. of order 6 l.cm (1, 6) 1 elementof z_4 and 6 elementof z_6 l.cm (6, 1) 6 Notelement z_4 and 2 elementof z_6 l.cm (2, 3) 2 elementof z_4 and 3 elementof z_6 l.cm (3, 2) 3 Notelement z_4 and 2 elementof z_6 l.cm (2, 6) 2 elementof z_4 and