Group
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Revision 245
7/7/2026, 8:54:01 AM · Ancient Tree
Concept edited
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1
A group is a [[Set|set]] $G$ equipped with a [[Binary operation|binary operation]] $*:G\times G \rightarrow G$ satisfying the following properties :2
1) Associativity : for all $a,b,c\in G$, $(a*b)*c=a*(b*c)$.3
2) Identity element : there exists an element $e\in G$ such that $e*a=a*e=a$ for every $a\in G$.4
3) Inverses : for every $a\in G$, there exists an element $a^{-1}\in G$ such that $a*a^{-1}=a^{-1}*a=e$.5
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##### Remarks7
- When the context is clear, the operation $*$ is often ommited, writing $ab$ for $a*b$.Revision 240
7/7/2026, 8:46:00 AM · Ancient Tree
Concept edited
Compare with revision 2393 changed lines
1
A group is a [[Set|set]] $G$ equipped with a [[Binary operation|binary operation]] $*:G\times G \rightarrow G$ satisfying the following properties :2
1) Associativity : for all $a,b,c\in G$, $(a*b)*c=a*(b*c)$.3
2) Identity element : there exists an element $e\in G$ such that $e*a=a*e=a$ for every $a\in G$.4
3) Inverses : for every $a\in G$, there exists an element $a^{-1}\in G$ such that $a*a^{-1}=a^{-1}*a=e$.5
6
##### Remarks7
- When the context is clear, the operation $*$ is often ommited, writing $ab$ for $a*b$.Revision 239
7/7/2026, 8:42:06 AM · Ancient Tree
Concept edited
Compare with revision 1095 changed lines
1
A group is...1
A group is a [[Set|set]] $G$ equipped with a [[Binary operation|binary operation]] $*:G\times G \rightarrow G$ satisfying the following properties :2
1) Associativity : for all $a,b,c\in G$, $(a*b)*c=a*(b*c)$.3
2) Identity element : there exists an element $e\in G$ such that $e*a=a*e=a$ for every $a\in G$.4
3) Inverses : for every $a\in G$, there exists an element $a^{-1}\in G$ such that $a*a^{-1}=a^{-1}*a=e$.Revision 109
6/27/2026, 8:24:10 AM · Ancient Tree
Concept created
A group is...