Nilpotent groups are solvable

I know this should be obvious but somehow I can’t seem to figure it out and it annoys me!
My definition of nilpotent groups is the following:
A group $G$ is nilpotent if every subgroup of $G$ is subnormal in $G$, or equivalently if $U<N_G(U)$ for all $U<G$. And my definition of solvable groups is that a group $G$ is solvable if $U’\neq U$ for all subgroups $1\neq U \le G$, where $U’$ is the commutator subgroup of $U$.
My question is then: why are nilpotent groups solvable?

Solutions Collecting From Web of "Nilpotent groups are solvable"

Let $G$ be nilpotent and nontrivial. Since every maximal subgroup of $G$ must be normal, and if $M$ is maximal then $G/M$ has no proper subgroups, it follows that if $M$ is maximal then $G/M$ is a group of order $p$, hence abelian. Therefore, $[G,G]\subseteq M$, since $[G,G]$ is contained in any normal subgroup $N$ of $G$ such that $G/N$ is abelian. In particular, $[G,G]\neq G$. Now simply note that being nilpotent is inherited to subgroups, as proven below, to conclude that $[H,H]\neq H$ for all subgroups $H$ of $G$ when $G$ is nilpotent. Hence, if every subgroup of $G$ is subnormal ($G$ is nilpotent), then the commutator subgroup of $H$ is properly contained in $H$ for any nontrivial subgroup $H$ of $G$ ($G$ is solvable).

(If $H\leq G$ and $K$ is a subgroup of $H$, then $K$ is subnormal in $G$, so there exist subgroups $K\triangleleft K_1\triangleleft K_2\triangleleft\cdots\triangleleft K_m=G$. Intersecting the subnormal series with $H$ gives you a subnormal series fo $K$ in $H$, showing $K$ is subnormal in $H$ as well; thus, every subgroup of $H$ is subnormal, so subgroup of nilpotent is nilpotent).

Added. I am tacitly assuming above that $G$ has maximal subgroups; so it might fail for infinite groups in which every subgroup is subnormal. In the infinite case, the usual definition of “nilpotent” is via either the upper central series or the lower central series, and that of “solvable” via the derived series. In the case of the definition via the lower central series, proving solvability is very easy: recall that the lower central series of $G$ is defined inductively by letting $G_1=G$ and $G_{n+1}=[G_n,G]$; and a group $G$ is nilpotent if and only if $G_{n+1}=\{1\}$ for some $n\geq 1$. Now note that $G^{(2)}=[G,G]=G_2$, and if $G^{(k)}\subseteq G_n$, then $G^{(k+1)} = [G^{(k)},G^{(k)}] \subseteq [G_k,G]=G_{k+1}$. So if the lower central series terminates, then so does the derived series, proving that if $G$ is nilpotent then $G$ is solvable.

Added: This follows exactly the same lines as Arturo’s answer above.


From your definitions, it is perhaps easiest to work with the maximal subgroup point of view on nilpotence.

To begin with, let $M$ be a maximal proper subgroup of $G$, which is nilpotent. Then (by your definition of nilpotence) we must have that $M$ is normal in $G$.
Thus $G/M$ is a group with no proper subgroups, hence is cyclic of prime order, and so in particular is abelian. Thus $G$ has a non-trivial abelian quotient, and so $G’$ is a proper subgroup of $G$.


Now we are done, provided that we can show that each subgroup of a nilpotent group is again nilpotent. This is not completely obvious using your definition, but is not too hard:

Suppose that $H$ is a subgroup of $G$, and that $U$ is a proper subgroup of $H$. By assumption $N_G(U)$ properly contains $U$. I claim that $N_H(U)$ (which equals $H \cap N_G(U)$) properly contains $U$ also. Indeed if not, i.e. if $N_H(U) = U$, then
$U =H \cap N_G(U)$. Now consider $N_G(N_G(U))$: we see that $$H \cap N_G(N_G(U)) = N_H(H \cap N_G(U)) = N_H(U) = U.$$ Continuing, we find
that $H \cap N_G(N_G( … N_G(U) …)) = U$. But since $N_G$ of any proper subgroup
properly contains that subgroup, the iterated $N_G$s on the left hand side eventually reach $G$,
and so we see that $H\cap G = U$. But $H \cap G = H$, and so $H = U$, contradicting our assumption that $U$ is a proper subgroup of $H$.
Thus indeed $N_H(U)$ properly contains $U$, and so $H$ is again nilpotent.