This chapter is a concise mathematical introduction into the algebra of groups. It is build up in the way that definitions are followed by propositions and proofs. The concepts and the terminology introduced here will serve as a basis for the following chapters that deal with group theory in the stricter sense and its application to problems in physics. The mathematical prerequisites are at the bachelor level. 1 This is an Open Access article distributed under the terms of the Creative Commons

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... ttribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. EPJ Web of Conferences Proof. a) Let e 1 and e 2 be two elements of (G, ) satisfying axiom 4. e 1 satisfies (4), so (e 1 , e 2 ) = e 2 . e 2 satisfies (4), so (e 1 , e 2 ) = e 1 . Therefore e 1 = (e 1 , e 2 ) = e 2 . b) Given a ∈ G and b 1 , b 2 two elements of (G, ) satisfying axiom 5. Then (a, b 1 ) = (b 1 , a) = e and (a, b 2 ) = (b 2 , a) = e. We have (b 1 , (a, b 2 )) = (b 1 , e) = b 1 . Because the law is associative (3), (b 1 , (a, b 2 )) = ( (b 1 , a), b 2 ) = (e, b 2 ) = b 2 , which means that b 1 = b 2 .