Infinite non-abelian $ p $-groups.

Question

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.

Answer 1

From the Wikipedia article on Nilpotent groups:

a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable.

see also these:

• O. Yu. Dashkova, Solvable infinite-dimensional linear groups with restrictions on the nonabelian subgroups of infinite rank, Siberian Math. J., Vol 49, No 6, pp 1023-1033, link
• S. N. Chernikov, Infinite nonabelian groups in which all infinite nonabelian subgroups are invariant, Ukrainian Math. J., Vol 23, No 5, pp 498-517, link

Answer 2

To answer your questions, yes every nilpotent group is solvable. That follows immediately from the definintions, because a finite central series for a group is also a normal series for which the factor groups are all abelian.

For the second question, it is not true that all infinite $p$-groups are nilpotent, or even solvable. The most extreme counterexample is a Tarski Monster, but there are easier couterexamples.

Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_{p^n}$ of degree $p^n$. Then, using a natural embedding $S_{p^n} \to S_{p^{n+1}}$, we get $P_n < P_{n+1}$, and the union $\cup_{n \ge 1} P_n$ is an infinite $p$-group. It cannot be solvable, because the derived length of $P_n$ tends to $\infty$ with $n$.