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

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Definition of a Group[edit]

Firstly, a Group is

a non-empty set, with a binary operation.[1]

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[edit]

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


a*b is in G if a, b are in Group G
(a*b)*c = a*(b*c)
if a, b, c are in Group G
1. Group G has an identity eG.
2. eG*c = c*eG = c if c is in Group G
1. if c is in G, c-1 is in G.
2. c*c-1 = c-1*c = eG


  1. Binary operation at wikipedia