I am working on a problem that requires taking quotients by normal subgroups that do not intersect a finite set of members of a group $G$. Is there a known collection of groups which always allows taking quotients by progressively smaller normal subgroups such that the normal subgroups do not intersect progressively larger finite set of group elements and the quotient is nilpotent?
I have come across the notion of residually nilpotent groups. Do they have this property? In particular here is the definition of residually nilpotent group. My question regarding them follows.
A group is residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent.
I am wondering if it is possible to escape a finite set of non-identity elements instead of just one. That is whether the following statement is true: if $G$ is residually nilpotent, then given any finite set of non-identity elements, there is a normal subgroup not containing those elements, such that the quotient group is nilpotent.
If not, then is there a collection of groups for which this statement is true? Is this collection of groups strictly larger than the collection of virtually nilpotent groups.
Best Answer
Suppose $G$ is residually nilpotent. Let $a_{1}, \dots, a_{n} \in G$. Then there are normal subgroups $N_{i}$, such that $a_{i} \notin N_{i}$ and $G/N_{i}$ is nilpotent.
Let $N = \bigcap \{ N_{i} : i = 1, \dots, n\}$. Then $a_{i} \notin N$ for all $i$, and $$ G/N \text{ is isomorphic to a subgroup of } \prod_{i=1}^{n} G/N_{i}, $$ the latter being nilpotent as a product of nilpotent groups.