Group and Field

Explore the abstract algebra objects group, abelian group, and field with examples.

Group

A group G, over a binary operation, oo, and two elements a,bGa, b \in G, is defined with the following four axioms:

  • Closure: A group is closed under oo, that is, the result of the operation is also a member of that group.

a o b=cGa\ o\ b = c \in G

  • Associativity: The result of a binary operation on three or more elements remains the same, regardless of the arrangement of parentheses.

(a o b) o c=a o (b o c), cG(a\ o\ b)\ o\ c = a\ o\ (b\ o\ c), \forall\ c \in G

  • Identity: There exists an identity element, ii, under operation, oo such that operation between any element of the group, cc, and ii results in cc
...