# Condition for Group given Semigroup with Idempotent Element

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## Theorem

Let $\struct {S, \circ}$ be a semigroup.

Let there exist an idempotent element $e$ of $S$ such that for all $a \in S$:

- there exists at least one element $x$ of $S$ satisfying $x \circ a = e$
- there exists at most one element $y$ of $S$ satisfying $a \circ y = e$.

Then $\struct {S, \circ}$ is a group.

## Proof

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## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.15$