Abstract Algebra/Group Theory/Group/Definition of a Group

= Definition of a Group =

Firstly, a Group is
 * a non-empty set, with a binary operation.

Secondly, if G is a Group, and the binary operation of Group G is $$\ast$$, then


 * 1. Closure
 * $$ \forall \; a, b \in G: a \ast b \in G $$
 * 2. Associativity
 * $$ \forall \; a, b, c \in G: (a \ast b) \ast c = a \ast (b \ast c) $$
 * 3. Identity
 * $$ \exists \; e_{G} \in G: \forall \; g \in G: e_{G} \ast g = g \ast e_{G} = g $$
 * 4. Inverse
 * $$ \forall \; g \in G: \exists \; g^{-1} \in G: g \ast g^{-1} = g^{-1} \ast g = e_{G} $$

From now on, eG always means identity of group G.

= Order of a Group =


 * Order of group G, o(G), is the number of distinct elements in G

= Diagram =

= References =