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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 20:25, 19 April 2012
, 20:25, 19 April 2012→6.6. Basic Notions of Group Theory: adjust table columns
A '''group''' is a monoid all of whose elements are invertible. That is, a group is a semigroup with a unit element in which every element has an inverse. Putting all the pieces together, then, a group <math>\underline{X} = (X, *, e)\!</math> is a set <math>X\!</math> with a binary operation <math>* : X \times X \to X\!</math> and a designated element <math>e\!</math> that is subject to the following three axioms:
A '''group''' is a monoid all of whose elements are invertible. That is, a group is a semigroup with a unit element in which every element has an inverse. Putting all the pieces together, then, a group <math>\underline{X} = (X, *, e)\!</math> is a set <math>X\!</math> with a binary operation <math>* : X \times X \to X\!</math> and a designated element <math>e\!</math> that is subject to the following three axioms:
{| align="center" width="90%"
{| align="center" cellspacing="10" width="90%"
| G1.
| width="5%" | '''G1.'''
| (associative)
| width="20%" | (associative)
| <math>x*(y*z) ~=~ (x*y)*z,\!</math>
| width="30%" | <math>x*(y*z) ~=~ (x*y)*z,\!</math>
| for all <math>x, y, z \in X.\!</math>
| width="45%" | for all <math>x, y, z \in X.\!</math>
|-
|-
| G2.
| '''G2.'''
| (identity)
| (identity)
| <math>e*x ~=~ x ~=~ x*e,\!</math>
| <math>e*x ~=~ x ~=~ x*e,\!</math>
| for some <math>e \in X.\!</math>
| for some <math>e \in X.\!</math>
|-
|-
| G3.
| '''G3.'''
| (inverses)
| (inverses)
| <math>x*y ~=~ e ~=~ y*x,\!</math>
| <math>x*y ~=~ e ~=~ y*x,\!</math>