I am trying to refresh my knowledge of modern algebra, and have come across a question in Gillian 8th ed. that I don't quite remember how to prove.

The question is:

For $n \geq 3$, describe the elements of $D_n$. How many elements does $D_n$ have?

I want to prove that $D_n$ is a complete set (I know it has $2n$ elements). I assume the general structure would be to prove that the list of symmetries (we'll call it $L$) is contained in the set of symmetries (we'll call it $\Sigma$), and then the other way, $\Sigma \subseteq L$.

If someone could point me in the right direction it would be much appreciated!


Hint Label the vertices $1,2,3,..,n$.

An element in $D_n$ will take $1$ to some vertex $k$ and $2$ to one of the two neighbours of $k$. Show that this completely determines the symmetry.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.