12,089
edits
Changes
Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 13:36, 20 April 2012
, 13:36, 20 April 2012→6.6. Basic Notions of Group Theory: small fixes
: In a semigroup <math>\underline{X} = (X, +, 0),\!</math> the <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> is notated as <math>nx\!</math> and defined for every positive integer <math>n\!</math> in the following manner. Proceeding recursively, let <math>1x = x\!</math> and let <math>nx = (n-1)x + x\!</math> for all <math>n > 1.\!</math>
: In a semigroup <math>\underline{X} = (X, +, 0),\!</math> the <math>n^\text{th}\!</math> '''multiple''' of an element <math>x\!</math> is notated as <math>nx\!</math> and defined for every positive integer <math>n\!</math> in the following manner. Proceeding recursively, let <math>1x = x\!</math> and let <math>nx = (n-1)x + x\!</math> for all <math>n > 1.\!</math>
: In a monoid <math>\underline{X} = (X, +, 0),\!</math> <math>nx\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>0x = 0\!</math> and proceeding the same way for <math>n > 0.\!</math>
: In a monoid <math>\underline{X} = (X, +, 0),\!</math> the multiple <math>nx\!</math> is defined for every non-negative integer <math>n\!</math> by letting <math>0x = 0\!</math> and proceeding the same way for <math>n > 0.\!</math>
: In a group <math>\underline{X} = (X, +, 0),\!</math> <math>nx\!</math> is defined for every integer <math>n\!</math> by letting <math>nx = (-n)(-x)\!</math> for <math>n < 0\!</math> and proceeding the same way for <math>n \ge 0.\!</math>
: In a group <math>\underline{X} = (X, +, 0),\!</math> the multiple <math>nx\!</math> is defined for every integer <math>n\!</math> by letting <math>nx = (-n)(-x)\!</math> for <math>n < 0\!</math> and proceeding the same way for <math>n \ge 0.\!</math>
: A group <math>\underline{X} = (X, +, 0),\!</math> is '''cyclic''' if and only if there is an element <math>g \in X\!</math> such that every <math>x \in X\!</math> can be written as <math>x = ng\!</math> for some <math>n \in \mathbb{Z}.\!</math> In this case, an element such as <math>g\!</math> is called a '''generator''' of the group.
: A group <math>\underline{X} = (X, +, 0)\!</math> is '''cyclic''' if and only if there is an element <math>g \in X\!</math> such that every <math>x \in X\!</math> can be written as <math>x = ng\!</math> for some <math>n \in \mathbb{Z}.\!</math> In this case, an element such as <math>g\!</math> is called a '''generator''' of the group.
<pre>
<pre>