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

Associativity:
(a*b)*c = a*(b*c)
if a, b, c are in Group G

# Definition of Associativity

Let G be a group with binary operation ${\displaystyle \ast }$

${\displaystyle \forall \;a,b,c\in G:(a\ast b)\ast c=a\ast (b\ast c)}$

# Usage

1. If a, b, c are in G, (a ${\displaystyle \ast }$ b) ${\displaystyle \ast }$ c = a ${\displaystyle \ast }$ (b ${\displaystyle \ast }$ c)

# Notice

1. G has to be a group
2. All of a, b and c have to be elements of G.
3. ${\displaystyle \ast }$ has to be the binary operation of G
4. The converse is not necessary true:
a (a ${\displaystyle \ast }$ b) ${\displaystyle \ast }$ c = a ${\displaystyle \ast }$ (b ${\displaystyle \ast }$ c) does not mean a, b or c must be in G.