Question

Question: Let G be a multiplicative group. Let $G^{op}$be the set G equipped with a new operation * defined by $a*b=ba$.

(a) Prove that$G^{op}$ is a group.

Short answer

It has been proved that $G^{op}$is a group.

Step-by-step solution

Step-By-Step Solution Step 1: show that  Gopis closed under *

Since G is a multiplicative group therefore, $ab\in G$for all $a,b\in G$.

Let $a,b\in G^{op}$then $ba\in G^{op}$.

Thus, $G^{op}$is closed under *.

Show that  Gopis associative

Let, $a,b,c\in G$

$$\begin{gathered} \left(a*b\right)*c=ba*c \\ =cba \\ =a*\left(cb\right) \\ =a*\left(b*c\right) \end{gathered}$$

Thus$G^{op}$is associative.

Show that  Gophas identity and inverse element

The identity element and inverse for G also works for $G^{op}$.

Conclusion

It can be concluded that $G^{op}$is a group.