This is concerning (part of) Exercise 5.4.5 and Exercise 5.4.6 of Robinson's, "A Course in the Theory of Groups (Second Edition)".
I have done the first one; the second might take me a while.
The Details:
The dihedral group is given by the presentation
$$D_{2n}\cong \langle r,s\mid r^n, s^2, srs=r^{-1}\rangle.$$
On page 150, paraphrased,
A group $G$ is supersolvable if it has a series of normal subgroups
$$1=G_0\unlhd G_1\unlhd \dots\unlhd G_n=G$$
all of whose factors $G_{i+1}/G_i$ are cyclic.
On page 157, we have the two exercises: part of
Exercise 5.4.5: The class of supersolvable groups is closed with respect to [. . .] finite direct products
. . . and . . .
Exercise 5.4.6: The product of two normal supersolvable subgroups need not be supersolvable. [Hint: Let $X$ be the subgroup of $GL(2,3)$ generated by
$$\begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}\text{ and }\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix},$$
thus $X\cong D_8$. Let $X$ act in the natural way on $A=\Bbb Z_3\oplus\Bbb Z_3$ and write $G=X\ltimes A$. Show that $G$ is not supersolvable. Let $L$ and $M$ be distinct Klein $4$-subgroups of $X$ and consider $H=LA$ and $K=MA$.]
The Question:
How does Exercise 5.4.5 not contradict Exercise 5.4.6?
Thoughts:
In Exercise 5.4.6, does Robinson mean the subgroup product
$$ST=\{ st\in C\mid s\in S, t\in T\}$$
for subgroups $S, T$ of a group $C$? I think so.
If not, then why does it not contradict Exercise 5.4.5?
A solution to (this part of) Exercise 5.4.5 can be found here:
Please help :)