Question

Let G=(Z/32Z)x

where x=multiplication and K=<[17]32>.
What is |G/K| and what are all the distinct closets of of K in G?

Answer 1

The index |G/K| is 1, indicating there is only one coset of K in G, which is G itself, because the subgroup K generated by [17]32 in (Z/32Z)× is of order 32.

Step-by-step explanation:

The question asks about finding the index of a subgroup K within a group G and identifying all the distinct cosets of K in G. Here, G is the group of integers modulo 32 under addition, and K is the subgroup generated by the equivalence class [17]32. The index of K in G, denoted |G/K|, is the number of distinct cosets of K in G.

Since 32 and 17 are coprime, the subgroup K generated by [17]32 in (Z/32Z)× is of order 32. Therefore, the index |G/K| is 1, and there is only a single coset which is G itself.

The distinct cosets of K in G can be represented as aK where a is an element of G, but since the index is 1, there is only one distinct coset, which is K itself.