Groups pp 1-37 | Cite as

Introductory Notions

  • Antonio Machì
Part of the UNITEXT book series (UNITEXT)


Definition 1.1. A group G is a non empty set in which it is defined a binary operation, i.e. a function:
$$ G\times G\to G $$
such that, if ab denotes the image of the pair (a, b),
  1. i)

    the operation is associative: (ab)c = a(bc), for all triples of elements a, b, cG;

  2. ii)

    there exists an element eG such that ea = a = ae, for all aG. This element is unique: if e′ is also such that e′a = a = ae′, for all aG, ea = a implies, with a = e′, that ee′ = e′, and a = ae′ implies, with a = e, that ee′ = e. Thus e′ = e;

  3. iii)

    for all aG, there exists bG such that ab = e = ba.



Finite Group Cyclic Group Proper Subgroup Dihedral Group Klein Group 
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Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  • Antonio Machì
    • 1
  1. 1.Department of MathematicsUniversity La SapienzaRomeItaly

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