Here is the question Let G be a group, H le G and N G. Denote delta H = {nh / n
ID: 2986378 • Letter: H
Question
Here is the question
Let G be a group, H le G and N G. Denote delta H = {nh / n N h H}. Prove NH G. Hint. Apply the subgroup criterion. Make use of the assumption, in particular N G. Consider the group (Z, +) and its normal subgroup 16Z = {16n|n Z} consisting all the multiples of 16. Denote the quotient group (Z/16Z,+) by (Z16, +) by (Z16 ,+);i.e., Z16 = z/16z. For every i Z, denote the coset i + 16Z by i-, which is an element in Z16. Determine o(6), the order of 6 in (Z/16Z.+). Determine [6], the cyclic subgroup generated by 6 in (Z16,+). Determine o(7), the order of 7 in (Z/16Z, +). Hint. The identity of (Z16, +) is 0. We write .& (3) Find the least n N such that n.i = 0. Determine [i] explicitly in the format of [I] = without repetition.Explanation / Answer
This is the continuation of the above answer
5.3)
1)order of 6' = O(6+16Z) = LCM(6,16)/6 = 8
2) [6'] is
{0+16Z,6+16Z, 12+16Z,18+16Z,24+16Z,30+16Z,36+16Z,42+16Z}
3) O(7') = LCM(7,16)/7 = 16