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Does there exist some sort of classification of subvarieties of $\mathfrak{N}_2$? Here $\mathfrak{N}_2$ stands for the variety of nilpotent groups of class $2$ (defined by the identity $[[x, y], z]$).

As we know that any variety is an intersection of a Burnside variety and a commutator variety (a variety in which all relatively free groups are torsion-free), it is sufficient to find all commutator subvarieties of $\mathfrak{N}_2$. Among them there are the variety of all abelian groups (defined by $[x, y]$ ). However, there definitely should be something else...

Chain Markov
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  • There is a geometric classification in low dimensions, i.e., up to $n\le 8$, see here for references (including my own :) ). – Dietrich Burde Jan 02 '20 at 16:05
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    I may be confused, but while the abelian groups are clearly a subvariety, I would think the 2-Engel groups aren't? '$[[x,y],z]$ is the identity for all $x,y,z$' is stronger than '$[[x,y],y]$ is the identity for all $x,y$' so there should be 2-Engel groups that aren't 2-nilpotent (and the groupprops wiki seems to agree; https://groupprops.subwiki.org/wiki/2-Engel_group ) – Steven Stadnicki Jan 02 '20 at 18:11
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    @StevenStadnicki: You are not confused; $\mathfrak{E}_2$ contains $\mathfrak{N}_2$, not the other way around. – Arturo Magidin Jan 02 '20 at 18:20

1 Answers1

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Yes, the full classification exists. It can be deduced from the classification of subvarieties of $\mathfrak{N}_3$ given in:

  1. Bjarni Jónsson, Varieties of groups of nilpotency three, Notices Amer. Math. Soc. 13 (1966), 488.

  2. V.N. Remeslennikov, Two remarks on 3-step nilpotent groups, Algebra i Logika (1965) no. 2, 59--65, MR 31:4838.

The first is an abstract, and has a typo (a $[y,x,z]$ should be $[y,x,x]$); the second contains the following classification:

Theorem. Every subvariety of $\mathfrak{N}_3$ corresponds to a $4$-tuple of nonnegative integers $(m,n,p,q)$ satisfying:

  1. $n|m/\gcd(2,m)$,
  2. $p|m$,
  3. $q|p$,
  4. $q|m/\gcd(6,m)$, and
  5. $p|3q$;

corresponding to the identities $$x^m = [x,y]^n = [x,y,z]^p = [x,y,y]^q = [x,y,z,w] = 1.$$

From this, by taking $p=q=1$, we get:

Corollary. Every subvariety of $\mathfrak{N}_2$ may be defined by the identities $$x^m=[x,y]^n=[x,y,z]=1$$ for unique nonnegative integers $n$ and $m$ satisfying $n|m/\gcd(2,m)$.

Note: Every group of nilpotency class at most two is $2$-Engel, but not every $2$-Engel group has nilpotency class at most two; the $2$-Engel groups are not a subvariety of $\mathfrak{N}_2$; the $2$-Engel groups are the groups whose $2$-generated subgroups are in $\mathfrak{N}_2$, which contains $\mathfrak{N}_2$. (The original post, prior to editing, included an off-hand assertion that the variety of $2$-Engel groups would be realized as a subvariety of $\mathfrak{N}_2$, hence this comment...)

Arturo Magidin
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