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Theorem: Construction of Groups from Commutative and Cancellative Semigroups
Let \((H,\circ)\) be a semigroup, which is commutative and cancellative. Then there exists a unique group \((G,\ast)\) with the following properties:
 There is a subset \(S\subset G\), which is a semigroup isomorphic to \(H\), i.e. where \((S,\ast)\simeq (H,\circ)\).
 There is a subset \(S\subset G\), which is a semigroup isomorphic to \(H\), i.e. where \((S,\ast)\simeq (H,\circ)\).
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Proofs: 2
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References
Bibliography
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013