This is Exercise 5.4.7 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE.
The Details:
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Denote the derived subgroup of a group $G$ by $G'$. It is the subgroup of $G$ generated by the commutators $[g,h]=g^{-1}h^{-1}gh$ of $G$ for all $g,h\in G$.
Since definitions vary, the definition of a normal subgroup $H\unlhd G$ of a group $G$ I use is a subgroup of $G$ such that $$aH=Ha$$ for all $a\in G$.
Robinson's definition of the following is equivalent to the one given on proof wiki:
Let $G$ be a group whose identity is $e$.
A normal series for $G$ is a sequence of normal subgroups of $G$:
$$\{e\}=G_0\lhd G_1\lhd\dots\lhd G_n=G,$$
where $G_{i-1}\lhd G_i$ denotes that $G_{i-1}$ is a proper normal subgroup of $G_i$.
On page 122 of Robinson's book,
Definition: A group $G$ is called nilpotent if it has a central series, that is, a normal series $1=G_0\le G_1\le \dots \le G_n=G$ such that $G_{i+1}/G_i$ is contained in the centre of $G/G_i$ for all $i$.
A supersolvable group $G$ is a group with a normal series
$$1=N_0\unlhd N_1\unlhd N_2\unlhd \dots\unlhd N_n=G$$
such that each factor $N_{i+1}/N_i$ is cyclic and $N_i\unlhd G$.
The Question:
Paraphrasing,
If $N\unlhd G$ is nilpotent and $G/N'$ is supersolvable, then $G$ is supersolvable.
Thoughts:
(This can be skipped.)
Usually, when stuck on a problem, I would try out an example; I couldn't think of a nilpotent subgroup $N\unlhd G$ such that $G/N'$ is supersolvable. I had a look here for examples of supersolvable groups, hoping to reverse engineer something to experiment with, but to no avail.
If it helps, I know that each supersolvable group is solvable (but the converse doesn't hold).
I can see roughly why the result holds. The quotient $G/N'$, being supersolvable, has a normal series whose quotients are cyclic, so we might be able to exploit one or more of the isomorphism theorems to get such a series for $G$; I don't see where $N$ being nilpotent is needed though.
I had a look in:
- Rotman's, "An Introduction to the Theory of Groups (Fourth Edition)".
- Roman's, "Fundamentals of Group Theory: An Advanced Approach".
- Hall's, "The Theory of Groups".
- Rose's, "A Course in Group Theory".
There is nothing obvious in them that I could find to help out here.
Here are some previous questions of mine on supersolvable groups:
- Find a property of groups that is inherited by quotients but not by subgroups.
- Two exercises by Robinson on supersolvable groups seem to contradict.
- If a chief factor of a group $G$ is not simple, then $G$ is not supersolvable. Is this true? If not, what is needed to make it true?
Previous questions of mine about nilpotent groups are listed here:
- If a nilpotent group has an element of prime order $p$, so does its centre.
- Show $G=\langle\delta\rangle\ltimes D$ is nilpotent of class $2$.
Please help :)