Find a group that contains elements $a$ and $b$ such that $|a|=2, |b|=11$ and $|ab|=2.$

Question

I was initially thinking along the lines of either finding some $S_n$ or some $D_n$ based on examples I saw here.

But the question is which of these $S_n$ or $D_n$ would be correct, and what would be the value of $n$.

Is $n = 11$, or is $n$ a multiple of $11$ and $2$?

Answer 1

$|a|=2$ means $a^2 =1$, so $a=a^{-1}$. Same for $|ab|=2$: $abab=aba^{-1}b=1$ so $aba^{-1}=b^{-1}$.

This is exactly (one of) the definition(s) of the dihedral group! Unluckily, you can find both notation $D_{11}$ and $D_{2 \times 11} = D_{22}$ in the literature for the* same group you are searching: check Wikipedia to understand what notation you are using.

*the dihedral group is the easiest example to find such $a,b$