definition of left or right coset of subgroup by element 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 definition of left or right coset of subgroup by element 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:
\( G'\): \(\in \{\text{ the groups }\}\)
\( G\): \(\in \{\text{ the subgroups of } G'\}\)
\( g'\): \(\in G'\)
\(*g' G\): \(= \text{ the left coset of } G \text{ by } g'\)
\(*G g'\): \(= \text{ the right coset of } G \text{ by } g'\)
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Conditions:
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2: Note
For any \(g'_1, g'_2 \in G'\), \(g'_1 G \cap g'_2 G = \emptyset\) or \(g'_1 G = g'_2 G\): when \(g'_1 G \cap g'_2 G \neq \emptyset\), there is a \(g'_3 \in g'_1 G \cap g'_2 G\), and \(g'_1 G = g'_3 G = g'_2 G\), 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.
Likewise, for any \(g'_1, g'_2 \in G'\), \(G g'_1 \cap G g'_2 = \emptyset\) or \(G g'_1 = G g'_2\).
\(\vert g' G \vert = \vert G \vert\): for each \(g_1, g_2 \in G\) such that \(g_1 \neq g_2\), \(g' g_1 \neq g' g_2\), because supposing that \(g' g_1 = g' g_2\), \(g_1 = {g'}^{-1} g' g_1 = {g'}^{-1} g' g_2 = g_2\), a contradiction.
Likewise, \(\vert G g' \vert = \vert G \vert\).
