Question

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

13. $\mathit{G}\mathbf{=}\mathbf{\mathbb{Z}}$;$\mathit{K}\mathbf{=}<7>$ (c)$\mathit{K}\mathbf{+}\left(-4\right)$ and $\mathit{K}\mathbf{+}\mathbf{59}$.

Short answer

The given cosets are identical.

Step-by-step solution

Step by Step Solution  Step 1: Subgroup K of G  

Given that K is a cyclic subgroup 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 $K+\left(-4\right)$consists of all integers leaving a remainder of -4 when divided by 7 and the coset can be written as follows:

$$\begin{gathered} K+59=K+\left(9·7-4\right) \\ =\left(K+9·7\right)-4 \\ =K-4 \end{gathered}$$

Since $-4\in K+59$implies that all the elements in $K+4$belong to the set $K+59$.

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

Therefore, the given cosets$K+4$and$K+59$are identical.