Question

Find the multiplicative inverse of each nonzero element in

(c)${\mathit{\mathbb{Z}}}_{\mathbf{7}}$

Short answer

It is a group of order $n$. It is abelian as well as cyclic in nature. If 1 is the identity element then it satisfies the relation${\mathit{a}}^{\mathbf{n}}\mathbf{=}\mathbf{1}$. It represents cyclicity and the multiplication table is given by:

Step-by-step solution

Definition of ℤn

It is a group of order $n$. It is abelian as well as cyclic in nature. If 1 is the identity element then it satisfies the relation${\mathit{a}}^{\mathbf{n}}\mathbf{=}\mathbf{1}$. It represents cyclicity and the multiplication table is given by:

Definition ofℤ5

It is a group of order 5. It is abelian as well as cyclic in nature. If 1 is the identity element then it satisfies the relation${\mathit{a}}^{\mathbf{5}}\mathbf{=}\mathbf{1}$. It represents cyclicity and the multiplication table is given by:

Inverse of an element

Inverse of an element $\mathit{a}$in a group $\mathit{G}$is defined as an element such that,

$\mathit{a}{\mathit{a}}^{\mathbf{\text{'}}}\mathbf{=}{\mathit{a}}^{\mathbf{\text{'}}}\mathit{a}\mathbf{=}\mathit{e}$where,$\mathit{e}$ is the identity element in$\mathit{G}$ .

Conclusion

By the table, we can see the non-zero elements are 1,2,3,4,5, and 6.

Their inverses are given by: