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Worksheet Coset Multiplication
To explore cosets a little more, let’s work with the group
\(G = S_3\) and the subgroup
\(H = \{(1), (23)\}\text{.}\)
1. Write down all the (left) cosets of
\(H\) in
\(G\text{.}\)
2.
Suppose we define an operation on the set of cosets using this rule:
\begin{equation*}
aH \star bH = (ab)H\text{.}
\end{equation*}
Find \((123)H\star (13)H\text{.}\) Now find \((123)H \star (132)H\text{.}\)
3. What if we switch our subgroup. Find the (left) cosets for
\(\hat H = \{(1), (123), (132)\}\text{.}\)
4. Using the same definition for
coset multiplication , find
\((123)\hat H\star (12)\hat H\) and
\((123)\hat H \star (13)\hat H\text{.}\) Do we run into the same concern?
5. Find the
right cosets for
\(H\) and
\(\hat H\text{.}\) What do you notice?
