Question

Find the multiplicative inverse of each nonzero element in

${\left(b\right)}^{{\mathbf{\mathbb{Z}}}_{\mathbf{5}}}$

Short answer

It is a group of order $\mathit{n}$. 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 $\mathit{n}$. 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 of ℤ5

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 a in group G is defined as an element a' such that,

aa'=a'a=e where, is the identity element in G.

Conclusion

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

Their inverses are given by: