Question

Suppose G is a cyclic group $<a>$ and $\left|a\right|$=15 . If role="math" localid="1651649969961" $\mathit{K}\mathbf{}\mathbf{=}<a^3>$, list all the distinct cosets of K in G.

Short answer

The distinct cosets of K in GareK e, K a, K a² .

Step-by-step solution

Group  G

Given that G is a cyclic group of order 15 which are {1, a ,a²,a³,a⁴,a⁵,a⁶,a⁷a,⁸,a⁹,a¹⁰,a¹¹,a¹²,a¹³,a¹⁴} .

Distinct cosets of  K in  G

Let K = be the subgroup of G then, the cosets of K can be separated as Ke{1, a³, ⁵,a⁶,a⁹,a¹²}, Ka² = {a²,a⁵,a,⁸,a¹¹,a¹⁴} . } and .

Therefore, the distinct cosets of K in G are Ke, Ka, Ka² .