Abelianise each of:
(a) $\Bbb Q \times S_4$
(b) $D_{12} \times A_4$
(c) $G \times Z_{10}$, where $G$ is the dicyclic group of order 12
and write down the torsion coefficients of the resulting abelian groups.
So I'm working on this question from Mark Armstrong's Groups and Symmetry, and I feel like I'm missing something. I know that "abelianization" is the process of making a non-abelian group abelian, and the way to do that is to find the commutator group and use it to divide the original group to get a quotient group that is abelian. I'm not sure how to go about this is a practical sense, though - to find the torsion coefficients, I needs groups of integers or something that is isomorphic to groups of integers, but I'm unsure of how to go from $\Bbb Q \times S_4$ to the quotient group I need
So, find the commutators and then mod out.
For the first one, $\Bbb Q$ is commutative. And it is well-known that the commutator of $S_n$ is $A_n$. So we get $\Bbb Q\times\Bbb Z_2$. And the torsion coefficient is $2$.
For the second, the commutators of the factors are $\langle r^2\rangle= D_{12}'$ and the klein four group $V= A_4'$. It is well known that the abelianization of $D_{2n}$ is the klein four group $V$. So, we get $V\times\Bbb Z_6$, and the torsion coefficients are $2,2$ and $6$.
Finally, the abelianization of the dicyclic group $G$ is known to be $C_4$, the commutator being $C_3$. We thus get $C_4\times\Bbb C_{10}\cong C_2\times C_{20}$, with torsion coefficients $2,20$.