Infinite nilpotent groups

Question

We know that all subgoups in a finite nilpotent groups are subnormal subgroup.

Is there exists an infinite nilpotent group, whose all subgroups are not subnormal subgroups?

Answer 1

No. Every subgroup of a nilpotent group is subnormal.

Proof: We prove by induction on the nilpotency class. Let $G$ be a nilpotent group. If $G$ is of nilpotency class $1$ then $G$ is abelian and hence every subgroup is normal. Suppose that every group of nilpotency class less than $n$ has every subgroup subnormal. Let $S$ be a a subgroup of $G$ and suppose that the nilpotency class of $G$ is $n$. Let $q:G\to G/Z(G)$ be the canonical quotient. Since $G/Z(G)$ is nilpotent with nilpotency class less than $n$ it follows that $q(S)$ is subnormal in $G/Z(G)$ and hence its inverse image $q^{-1}(q(S)) = Z(G)S$ is subnormal in $G$. Since $S$ is normal in $Z(G)S$ it follows that $S$ is subnormal in $G$.

Note 1: One can easily adapt the above to show that: every subgroup of a group of nilpotency class $n$ is subnormal of depth at most $n$.

Note 2: We could have taken as "base case" nilpotency class $0$ i.e.when $G$ is trivial.