A group is a mathematical construct having closure, inverse, identity and Associativity.
I am having trouble in proving (or disproving) if the inverse of an element are unique?
I have tried to prove $b=c$ in the following cases:
Here $a$ and $b$ belong to the set and $\cdot$ is used to show the operation
1.
$$a\cdot b=e_1$$
$$a\cdot c=e_1$$
where $x\cdot e_1=x$ (right identity, i.e., $e_1$ is multiplied to the right hand side to get $x$)
- $$a\cdot b=e_2$$
$$a\cdot c=e_2$$
$e_2\cdot x=x$ (left identity, i.e., $e_2$ is multiplied to the left hand side to get $x$)
In which case and why, we can say that $b=c$?
Best Answer
The trick here is to pit two inverses $s,t$ of an element $x$ against each other. Let $e$ be the identity element.
By definition, $s\cdot x=e=x\cdot s$ and $t\cdot x=e=x\cdot t$.
We have
$$\begin{align} s&=s\cdot e\\ &=s\cdot (x\cdot t)\\ &=(s\cdot x)\cdot t\\ &=e\cdot t\\ &=t. \end{align}$$