A description/proof of that for coset map with respect to subgroup, preimage of image of subset is subgroup multiplied by subset
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of left or right coset of subgroup by element of group.
- The reader admits the proposition that with respect to any subgroup, the coset by any element of the group equals a coset if and only if the element is a member of the latter coset, whether they are left cosets or right cosets.
Target Context
- The reader will have a description and a proof of the proposition that for the left or right coset map with respect to any subgroup, the preimage of the image of any subset is the subgroup multiplied by the subset from left or right, respectively.
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: Description
For any group, \(G_1\), any subgroup, \(G_2 \subseteq G_1\), the left or right coset map, \(\pi: G_1 \rightarrow G_1/G_2\), and any subset, \(S \subseteq G_1\), \(\pi^{-1} \pi (S) = S G_2\) or \(\pi^{-1} \pi (S) = G_2 S\).
2: Proof
Let us prove it for the left coset map. For any element, \(p \in \pi^{-1} \pi (S)\), \(\pi (p) \in \pi (S)\). There is an element, \(p' \in S\), such that \(\pi (p) = \pi (p')\), which means that \(p G_2 = p' G_2\). So, \(p \in p' G_2\), so, \(p \in S G_2\). For any element, \(p \in S G_2\), \(p \in p' G_2\) for an element, \(p' \in S\). By the proposition that with respect to any subgroup, the coset by any element of the group equals a coset if and only if the element is a member of the latter coset, whether they are left cosets or right cosets, \(p G_2 = p' G_2\). \(\pi (p) \in \pi (S)\). \(p \in \pi^{-1} \pi (S)\).
Let us prove it for the right coset map. For any element, \(p \in \pi^{-1} \pi (S)\), \(\pi (p) \in \pi (S)\). There is an element, \(p' \in S\), such that \(\pi (p) = \pi (p')\), which means that \(G_2 p = G_2 p'\). So, \(p \in G_2 p'\), so, \(p \in G_2 S\). For any element, \(p \in G_2 S\), \(p \in G_2 p'\) for an element, \(p' \in S\). By the proposition that with respect to any subgroup, the coset by any element of the group equals a coset if and only if the element is a member of the latter coset, whether they are left cosets or right cosets, \(G_2 p = G_2 p'\). \(\pi (p) \in \pi (S)\). \(p \in \pi^{-1} \pi (S)\).
