Let there be dihedral group $D_n$ and take two elements with first ($a$) being a generator rotation and other ($b$) any reflection. Which group is $[D_n,D_n]$?
Till now not covered quotient groups, normal groups, or conjugates even.
Request vetting of understanding of the problem :
Problem:
I have some idea about commutator as:
Given $a,b\in G$ for a group $G$, define the commutator of group element/composition $ab$ to be $[a,b]=aba^{-1}b^{-1}= ab(ba)^{-1}$.
So, do need to check if each element $ab, (ba)^{-1}$ is a generator of the whole group $D_n$?
But, here states:
If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by commutator elements from $X$ and $Y$ is $[X, Y] = \langle [x, y] \mid x\in X, y\in Y\rangle$.
By it, that means that:
Need to check if each element $a,b$ (in $ab$) is a generator of the whole group $D_n$?
The first interpretation seemed intuitive, as each of $ab, (ba)^{-1}$ is an element as well as an entry in the group table.
So, taking the two elements as seperately involved in the composition $ab$ didn't seem a natural way to interpret.
If am wrong (second interpretation is correct), then please tell.