View source for Abstract Algebra/Group Theory/Group/Definition of a Group/Definition of Associativity

[[File:Algebra_Proof_Diagram_Associativity.svg‎|frame| ''Associativity'': <br/>(a*b)*c = a*(b*c) <br /> if a, b, c are in Group G]]

=Definition of Associativity=
Let G be a [[../|group]] with [[../../../../Binary Operations|binary operation]] <math>\ast</math>

:<math> \forall \; a, b, c \in G: (a \ast b) \ast c = a \ast (b \ast c)  </math>

=Usage=
# {{Anchor|Usage1}}If ''a'', ''b'', ''c'' are in G, (''a <math>\ast</math> ''b'') <math>\ast</math> ''c'' = ''a <math>\ast</math> (''b'' <math>\ast</math> ''c'')

=Notice=
# G has to be a [[../|group]]
# All of ''a'', ''b'' and ''c'' have to be elements of G.
# <math>\ast</math> has to be the binary operation of G
# The converse is not necessary true:
::  ''a'' (''a <math>\ast</math> ''b'') <math>\ast</math> ''c'' = ''a <math>\ast</math> (''b'' <math>\ast</math> ''c'') does not mean ''a'', ''b'' or ''c'' must be in G.

{{BookCat}}
<!--These empty lines allow links to anchor on this page go right to the anchor-->

<!--These empty lines allow links to anchor on this page go right to the anchor-->