Existence of an identity element in a group

#group_theory

Suppose $G$ is a nonempty finite set that has an associative pairing $G \times G \to G$ , written $(x, y) \mapsto x \cdot y$ . Suppose this pairing satisfies left and right cancellation: $x \cdot y = x \cdot y^{'}$ implies $y = y^{'}$ , and $x \cdot y = x^{'} \cdot y$ implies $x = x^{'}$ . Prove there exists an element $e$ of $G$ such that for all $x \in G$ , $e \cdot x = x \cdot e = x$ . Justify your reasoning as carefully as possible.

View

L A T E X

code

Suppose $G$ is a nonempty finite set that has an associative pairing $G\times G\to G$, written $(x,y)\mapsto x\cdot y$. Suppose this pairing satisfies left and right cancellation: $x\cdot y = x\cdot y'$ implies $y=y'$, and $x\cdot y = x'\cdot y$ implies $x=x'$. Prove there exists an element $e$ of $G$ such that for all $x\in G$, $e\cdot x = x\cdot e = x$. Justify your reasoning as carefully as possible.