MathLiveGroup axioms
These conditions define a group: a set with a binary operation that is closed, associative, has an identity element, and in which every element has an inverse. They are the starting point for all of group theory.
\begin{aligned}&(\cdot)\colon G\times G\to G\\&\forall x,y,z\in G\colon (x\cdot y)\cdot z=x\cdot(y\cdot z)\\&\exists1\in G\forall x\in G\colon 1\cdot x=x\cdot1=x\\&\forall x\in G\exists y\in G\colon x\cdot y=y\cdot x=1\end{aligned}What the symbols mean
| the set forming the group | |
| the binary group operation mapping G\times G to G | |
| arbitrary elements of G | |
| the identity element of G |
About this formula
- FieldMathematics
- TopicAbstract algebra
- LevelUniversity
- Also calledgroup definition, definition of a group, associativity identity inverse, mathematical group, group (algebra), group (math)
- SourceWikidata Q83478
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