description/proof of that intersection of subgroups of group is subgroup of group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of group.
Target Context
- The reader will have a description and a proof of the proposition that for any group, the intersection of any possibly uncountable number of subgroups of the group is a subgroup of the 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:
\(G'\): \(\in \{\text{ the groups }\}\)
\(S\): \(\subseteq \{\text{ the subgroups of } G'\}\)
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Statements:
\(\cap S \in \{\text{ the subgroups of } G'\}\)
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2: Proof
Whole Strategy: Step 1: see that \(\cap S\) satisfies the requirements to be a group.
Step 1:
For the identity element, \(1 \in G'\), \(1 \in \cap S\), because for each \(G \in S\), \(1 \in G\).
For any \(g_1, g_2 \in \cap S\), \(g_1 g_2 \in \cap S\), because for each \(G \in S\), \(g_1, g_2 \in G\), and so, for each \(G \in S\), \(g_1 g_2 \in G\).
For any \(g \in \cap S\), \(g^{-1} \in \cap S\), because for each \(G \in S\), \(g \in G\), and so, for each \(G \in S\), \(g^{-1} \in G\).
The associativity of multiplications holds, because it holds in the ambient \(G'\).
