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group (algebra)

1 definition: group

A non-empty set \(G\) is a group if it has a defined operation \((\cdot)\) such that:

  1. (closure) \(a,b\in G \Rightarrow a\cdot b \in G\)
  2. (associative law) If \(a,b,c\in G\), then \(a\cdot (b \cdot c) = (a \cdot b) \cdot c\)
  3. (identity element) There exists an \(e\in G\) such that \(a \cdot e = e \cdot a = a\) for all \(a \in G\)
  4. (inverse) For every \(a\in G\), there exists \(a^{-1}\in G\) such that \(a\cdot a^{-1} = a^{-1}\cdot a = e\)

1.1 related algebraic structures

1.1.1 definition: semi-group

A group that does not necessarily have an identity element or an inverse for every element.

1.1.2 definition: monoid

A group that does not necessarily have an inverse for every element.

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Created: 2021-09-14 Tue 21:44