Question

Question:In Exercise 13-15, K is a subgroup of G. Determine whether the given cosets are disjoint or identical.

13. $G=\mathrm{\mathbb{Z}}$; $\mathit{K}\mathbf{=}<7>$(b)K=4 andk+137 .

Short answer

The given cosets are identical.

Step-by-step solution

Step by Step Solution step 1 :dubgroup K of G  

Given that K is a cyclic subgroup (7) of the additive group Z .

The subgroup K consists of all the multiples of 7 and the cosets of K are the congruence classes modulo 7.

The cosets are identical 

The coset $K+4$ consists of all integers leaving a remainder 4 when divided by 7 and the coset $K+137$can be written as follows:

$$\begin{gathered} K+137=K+\left(19·7+4\right) \\ =\left(K+19·7\right)+4 \\ =K+4 \end{gathered}$$

Since,$4\in K+137$implies that the elements in $K+4$belongs to set $K+137$.

Thus, the intersection of two cosets are non-empty.

Therefore, the given cosets K+4 andK+137 are identical.