To find the distinct right cosets of k in g, we can use the formula:
gk ={g * kI g є G}
First, let's define the group G and the subgroup K given in the question:
G = U32 (the set of units modulo 32)
K = {1, 17}
To find the right cosets, we need to multiply each element of G by each element of K. Since G and K have finite sets, we can iterate through their elements.
Here are the distinct right cosets of K in G:
1. Coset 1: g *1=g
For each element g in G, the coset is {g).
Example: (1, 3, 5, 7, 9, 11, 13, 15, 17,
19, 21, 23, 25, 27, 29, 313
2. Coset 17:
g * 17 = g * (1 + 16) = g + 16
For each element g in G, the coset is {g + 16}.
Example: {17, 19, 21, 23, 25, 27, 29,
31, 1, 3, 5, 7, 9, 11, 13, 15}
Note that these cosets are distinct and do not overlap. Each coset represents a set of elements obtained by multiplying each element in G by a fixed element in K.
By following these steps, you can find the distinct right cosets of a subgroup in a group.