Definition#

A group is a set \(G\), together with a binary operator \(\star\), satisfying the following conditions:

  1. \(a \star b \in G\) for all \(a,b \in G\) (closure).
  2. \((a \star b) \star c = a \star (b \star c)\) for all \(a,b,c \in G\) (associativity).
  3. There exists an \(e \in G\) such that \(a \star e = e \star a = a\) for all \(a \in G\) (existence of identity).
  4. For each \(a \in G\), there exists a \(b \in G\) such that \(a \star b = b \star a = e\) (existence of inverses).

Example#

See this exercise.

Source#

Gregory T. Lee. Abstract Algebra. 1st ed. Springer Cham, 2018. pp. 38.