definition of general linear group of module
Topics
About: module
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% module.
- The reader knows a definition of group.
- The reader knows a definition of linear map.
- The reader knows a definition of bijection.
Target Context
- The reader will have a definition of general linear group of module.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\( M\): \(\in \{\text{ the } R \text{ modules }\}\)
\(*GL (M)\): \(= \{f: M \to M \vert f \in \{\text{ the linear maps }\} \cap \{\text{ the bijections }\}\}\), \(\in \{\text{ the groups }\}\), with the group operation, \(g\), specified below
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Conditions:
\(g: GL (M) \times GL (M) \to GL (M), (f_1, f_2) \mapsto f_1 \circ f_2\)
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2: Note
It is indeed a group, because for any elements, \(f_1, f_2, f_3 \in GL (M)\), 1) \((f_1 \bullet f_2) \bullet f_3 = f_1 \bullet (f_2 \bullet f_3)\), by Note for the definition of composition of maps; 2) the identity map, \(id: M \to M\), is in \(GL (M)\) and is the identity element, because \(id \bullet f_1 = f_1 \bullet id = f_1\); 3) \(f_1\) has the inverse, \(f_1^{-1}: M \to M\), by the proposition that any bijective linear map between any modules is a 'modules - linear morphisms' isomorphism, which is in \(GL (M)\), because it is linear and bijective, because it has the inverse, \(f_1\), and the proposition that any map is a bijection if and only if it has an inverse applies.
While general linear group of vectors space has been defined, in fact, the vectors space does not really need to be a vectors space but can be a module.
