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Difference between revisions of "Abelian group"
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An '''abelian group''' is a [[group]] in which the group [[operation]] is [[commutative]].
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An '''abelian group''' is a [[group]] in which the group [[operation]] is [[commutative]]. They are named after Norwegian mathematician Niels Abel.  
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For a [[group]] to be considered "abelian", it must meet several requirements.
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For a [[group]] to be considered '''abelian''', it must meet several requirements.
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"Closure"
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           For all <math>a,b</math> <math>/in</math> <math>S</math>, and for all functions <math>/bullet</math>, <math>a/bullet b /in S</math>.
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Closure
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           For all <math>a,b</math> <math>\in</math> <math>S</math>, and for all operations <math>\bullet</math>, <math>a\bullet b \in S</math>.
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Associativity
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          For all <math>a,b,c</math> <math>\in</math> <math>S</math> and all operations <math>\bullet</math>, <math>(a\bullet b)\bullet c=a\bullet(b\bullet c)</math>.
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Identity Element
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          There exists some <math>e \in S</math> such that <math>a \bullet e = e \bullet a = a</math>.
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Inverse Element
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          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
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          For all <math>a,b \in S</math>, <math>a \bullet b = b \bullet a</math>.
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A simple example of an abelian group is <math>\mathbb{Z}</math> under addition. It is simple to show that it meets all the requirements.
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Closure
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          For all <math>a,b \in \mathbb{Z} , a+b \in \mathbb{Z}</math>.
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Associativity
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          For all <math>a,b,c \in \mathbb{Z} , (a+b)+c = a+(b+c)</math>.
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Identity Element
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          For all <math>a \in \mathbb{Z} , a+0 = 0+a = a</math>.
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Inverse Element
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          For all <math>a \in \mathbb{Z} , a+ -a = 0</math>.
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Commutativity
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          For all <math>a,b \in \mathbb{Z} , a+b = b+a</math>.
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Seeing as <math>\mathbb{Z}</math> meets all of these requirements under addition, we can say that <math>\mathbb{Z}</math> is abelian under addition.
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==Examples==
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Notable examples of abelian groups include the integers under addition, the real numbers under addition, the integers modulo <math>n</math> under addition, the multiplicative group of integers modulo <math>n</math>, and the additive group of any ring. Many matrix groups are ''not'' abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6.  
  
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{{stub}}
 
  
 
[[Category:Definition]]
 
[[Category:Definition]]
 
[[Category:Group theory]]
 
[[Category:Group theory]]

Latest revision as of 17:30, 14 June 2020

An abelian group is a group in which the group operation is commutative. They are named after Norwegian mathematician Niels Abel. 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 $a,b \in S$, $a \bullet b = b \bullet a$.

A simple example of an abelian group is $\mathbb{Z}$ under addition. It is simple to show that it meets all the requirements.

Closure 

         For all $a,b \in \mathbb{Z} , a+b \in \mathbb{Z}$.

Associativity 

         For all $a,b,c \in \mathbb{Z} , (a+b)+c = a+(b+c)$.

Identity Element 

         For all $a \in \mathbb{Z} , a+0 = 0+a = a$.

Inverse Element 

         For all $a \in \mathbb{Z} , a+ -a = 0$.

Commutativity 

         For all $a,b \in \mathbb{Z} , a+b = b+a$.

Seeing as $\mathbb{Z}$ meets all of these requirements under addition, we can say that $\mathbb{Z}$ is abelian under addition.

Examples

Notable examples of abelian groups include the integers under addition, the real numbers under addition, the integers modulo $n$ under addition, the multiplicative group of integers modulo $n$, and the additive group of any ring. Many matrix groups are not abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6.

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