Find a subgroup $H$ of $S_6$ such that $D_3$ is isomorphic to $H$
I really need help in this question perparing for a test.
Find a subgroup $H$ of $S_6$ such that $D_3$ is isomorphic to $H$
I really need help in this question perparing for a test.
Another approach to this problem is to note that $D_3$ is a group of six elements. Therefore the group action of $D_3$ on itself by (say) left multiplication is a mapping of $D_3$ to permutations of six things $S_6$.
If you know that in fact $\,D_3\cong S_3\,$ then all you need to do is to choose a subgroup of $\,S_6\,$ that only "moves" three elements.
For example, taking $\,S_n\,$ as the group of permutations on $\,\{1,2,3,...,n\}\,$, you can take $H:=\{\sigma\in S_6\;:\;\sigma(i)=i\,\,\,\forall \,i=4,5,6\}$
Hint: $D_3$ is generated by an element of order $3$ and an element of order $2$. Try finding an element of order $3$ and an element of order $2$ in $S_6$, and let $H$ be the subgroup generated by them.