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Difference between revisions of "Abelian group"
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Inverse Element
 
Inverse Element
 
           For all <math>a \in S</math>, there exists some <math>a^{-1}</math> such that <math>a \bullet a^{-1} = e</math>
 
           For all <math>a \in S</math>, there exists some <math>a^{-1}</math> such that <math>a \bullet a^{-1} = e</math>
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+
Commutativity
 +
          For all <math>a,b \in S</math>, <math>a \bullet b = b \bullet a</math>.
  
 
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Revision as of 17:42, 12 August 2015

An abelian group is a group in which the group operation is commutative. For a group to be considered "abelian", it must meet several requirements.

Closure 

         For all $a,b$ $\in$ $S$, and for all operations $\bullet$, $a\bullet b \in S$.

Associativity 

         For all $a,b,c$ $\in$ $S$ and all operations $\bullet$, $(a\bullet b)\bullet c=a\bullet(b\bullet c)$.

Identity Element 

         There exists some $e \in S$ such that $a \bullet e = e \bullet a = a$.

Inverse Element 

         For all $a \in S$, there exists some $a^{-1}$ such that $a \bullet a^{-1} = e$

Commutativity 

         For all $a,b \in S$, $a \bullet b = b \bullet a$.

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