Question

If $a,b\in G$and, $ab=e$, prove that $ba=e$

Short answer

It is proved that $ba=e$.

Step-by-step solution

Identity element of a group

The identity element in a group is defined as an elementsuch$e\in G$ that

$e\ast a=a\ast e=a$for all$a\in G$

Inverse of an element

Let be a group. An element $a^{-1}\in G$is said to be the inverse of an element $a\in G$if it satisfies, $a\ast a^{-1}=a^{-1}\ast a=e$where $e\in G$is the identity element.

Proof

Given that, $ab=e$Multiplying both sides by the inverse of from the left, we get,

$\begin{array}{l}{a}^{-1}\left(ab\right)=a^{-1}e\\ \Rightarrow \left(a^{-1}a\right)b=a^{-1}\cdot e\\ \Rightarrow e\cdot b=a^{-1}\cdot e\\ \Rightarrow b=a^{-1}\cdot e=e\cdot a^{-1}\\ \Rightarrow b=e\cdot a^{-1}\end{array}$

Now, multiplying both sides by from the right,

$\begin{array}{l}ba=\left(e{a}^{-1}\right)a\\ =e\left(a^{-1}a\right)\\ =e\cdot e\\ =e\end{array}$

Hence, it is proved that $ba=e$.