I read somewhere that Maltcev proved that for any finitely generated torsion free Nilpotent group $G$ there are canonical generators, i.e. $g_{1},\ldots,g_{k}$ such that any $g \in G$ can be written uniquely as: $g_{1}^{a_{1}}g_{2}^{a_{2}} \cdots g_{k}^{a_{k}}$, where the $a$-s are in $Z$. I could not find the paper online, probably since it is very old.

I have several questions (2-4 are versions of the same question):

1) If the size of the set of canonical generators is $k$ does this implies that the group has polynomial growth with exponent $k$?

2) Can we always pick one of the canonical generators to be in the center of the group?

3) If 2 fails, is there always a subgroup of finite index for which 2 is true?

4) I think this is can imply 3: can we pick canonical generators, $g_{1},\ldots,g_{k}$ and $z$ in the center, such that if we throw away one of the canonical generators, the rest of the canonical generators do not have z in the group they generate?

The motivation is: consider a general group G of polynomial growth with exponent d>2. I want to be to find a copy of $Z^{2}$, call it M, inside G and a group of polynomial growth with exponent d-2, call it H, such that if we look at $mH$ are disjoint for different $m-s \in M$. For some application I want to treat the $mH$-s as hyperplanes and it is important that I can pick them to be disjoint.