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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 19:44, 19 April 2012
, 19:44, 19 April 2012→6.6. Basic Notions of Group Theory: markup
: <math>y' ~=~ y'*e ~=~ y'*(x*y) ~=~ (y'*x)*y ~=~ e*y ~=~ y.\!</math>
: <math>y' ~=~ y'*e ~=~ y'*(x*y) ~=~ (y'*x)*y ~=~ e*y ~=~ y.\!</math>
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 X = <X, *, e> is a set X with a binary operation * : XxX >X and a designated element e that is subject to the following three axioms:
G1. (associative) x*(y*z) = (x*y)*z, for all x, y, z C X.
{| align="center" width="90%"
| G1.
G2. (identity) e*x = x = x*e, for some e C X.
| (associative)
| <math>x*(y*z) ~=~ (x*y)*z,\!</math>
G3. (inverses) x*y = e = y*x, for some y C X,
| for all <math>x, y, z \in X.\!</math>
|-
| G2.
| (identity)
| <math>e*x ~=~ x ~=~ x*e,\!</math>
| for some <math>e \in X.\!</math>
|-
| G3.
| (inverses)
| <math>x*y ~=~ e ~=~ y*x,\!</math>
| for some <math>y \in X,\!</math> for all <math>x \in X.\!</math>
|}
<pre>
It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups. A system X = <X, *> is given the adjective "commutative" if and only if * is commutative. Commutative groups, however, are traditionally called "abelian groups". By way of making comparisons with familiar systems and operations, the following usages are also common.
It is customary to use a number of abbreviations and conventions in discussing semigroups, monoids, and groups. A system X = <X, *> is given the adjective "commutative" if and only if * is commutative. Commutative groups, however, are traditionally called "abelian groups". By way of making comparisons with familiar systems and operations, the following usages are also common.