Find all subgroups of cyclic group - find all subgroups of

Question

1 (i) Let C_n = {e, x, ..., x^n-1} (xⁿ = e) be the cyclic group of order n and let H be a subgroup of C_n . By considering min{i > 0 : xⁱ ∈ H} show that H = (x^m) for some m|n.

(ii) Find all subgroups of C₁₈, explaining your answer.

2(i) Find (a) 4¹⁶⁰² (mod 17), (b) 4^-1 (mod 17), (c) 4¹⁵⁹⁷ (mod 17)

(ii) Find (a) 11^-1 (mod 13), (b) 11¹¹⁹⁸ (mod 13)

Attachment:- Maths-Groups.pdf