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Difference between revisions of "Abelian group"
Line 9: Line 9:
 
           There exists some <math>e \in S</math> such that <math>a \bullet e = e \bullet a = a</math>.
 
           There exists some <math>e \in S</math> such that <math>a \bullet e = e \bullet a = a</math>.
 
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>
  
  

Revision as of 17:41, 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 $a \bullet a^{-1} = e$


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