definition of subgroup generated by subset of group
Topics
About: group
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of group.
Target Context
- The reader will have a definition of subgroup generated by subset of group.
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:
\( G'\): \(\in \{\text{ the groups }\}\)
\( S\): \(\subseteq G'\)
\(*(S)\): \(\in \{\text{ the subgroups of } G'\}\)
//
Conditions:
\(S \subseteq (S)\)
\(\land\)
\(\lnot \exists G \in \{\text{ the subgroups of } G'\} (S \subseteq G \land G \subset (S))\)
//
In other words, \((S)\) is the smallest subgroup that contains \(S\).
2: Natural Language Description
For any group, \(G'\), and any subset, \(S \subseteq G'\), the smallest subgroup that contains \(S\), \((S)\)
3: Note
\((S)\) is indeed well-defined (uniquely exists), because it is the intersection of all the subgroups that contain \(S\): there is at least 1 subgroup that contains \(S\), \(G'\); the intersection is indeed a smallest subgroup that contains \(S\), because it contains \(S\), the intersection of any subgroups is a subgroup, and it is a smallest, because it is contained in any subgroup that contains \(S\); the intersection is indeed the unique smallest subgroup that contains \(S\), because if there was another smallest subgroup that contains \(S\), it would be contained in the constituents of the intersection, and the intersection would be smaller than or equal to that another smallest subgroup, and if the intersection was smaller than that another smallest subgroup, that another smallest subgroup was not in fact smallest, and if the intersection was equal to that another smallest subgroup, that another smallest subgroup was not in fact "another".
When \(S \neq \emptyset\), \((S)\) is the set of all the finite multiplications of the elements of \(S\) and their inverses, which can be another definition of subgroup generated by subset of group.
That is because the set contains \(S\); each element of the set is contained in \((S)\), because \((S)\) is a group; the set indeed constitutes a group, because it is closed in multiplications, \(g g^{-1} = 1\) is contained in it, for each \(g_1 ... g_k\) in the set, the inverse, \(g_k^{-1} ... g_1^{-1}\), is in the set, and so, the set is a subgroup that contains \(S\) that (the set) is smaller than or equal to \((S)\), but the set cannot be smaller because \((S)\) is the smallest, so, the set is equal to \((S)\).
When \(S = \emptyset\), \((S) = \{1\}\).
When \(S = \{g\}\), \((S)\) is usually denoted as \((g)\).
