Lately, I have been constructing finite involution monoids that generate varieties with $2^{\aleph\_0}$ subvarieties. One construction requires groups that violate the identity ${ [x,y]^2 \approx 1 }$, where ${ [x,y] = x^{-1} y^{-1} xy }$. 


Is there a name for groups satisfying the identity ${ [x,y]^2 \approx 1 }$? Has there been any work done on these groups?



Groups that satisfy ${ [x,y]^2 \approx 1 }$
This variety of groups has indeed been considered in the literature. It is known that the following conditions hold for every group $G$ satisfying the identity $[x,y]^2=1$:


1. $[[x,y\_1,\ldots,y\_m],[x,z\_1,\ldots,z\_n]]=1$ for all
$x,y\_1,\ldots,y\_m,z\_1,\ldots,z\_n\in G$ (see [1]).
2. $[[x\_1,x\_2],[x\_3,x\_4]]=[[x\_{\pi(1)},x\_{\pi(2)}],[x\_{\pi(3)},x\_{\pi(4)}]]$ for all $x\_1,\ldots,x\_4\in G$ and $\pi\in S\_4$ (see [2]).
3. $[[x,y],[z,w]]=[x,y,z,w][x,y,w,z]$ (see [1]).
4. $[\gamma\_2(G),\gamma\_3(G)]=1$ (see [2]).
5. $[G'',G]=1$ (see [2]).
6. $G'^4=1$ (see [2]).
7. The group $G$ need not be metabelian (see [3]).
8. If $G$ is finite, then $G=P\rtimes H$, where $P$ is a normal Sylow $2$-subgroup of $G$, $H$ is abelian of odd order, and $[P,H]$ is an elementary abelian $2$-group (see [1]).


**References:**


1. M. Farrokhi D. G., [On groups satisfying a symmetric Engel word](http://dx.doi.org/10.1007/s11587-015-0239-7), *Ric. Mat.* **65** (2016), 15–20.
2. I. D. Macdonald, [On certain varieties of groups](http://dx.doi.org/10.1007/BF01210977), *Math. Z.* **76**, (1961) 270–282.
3. B. H. Neumann, [On a conjecture of Hanna Neumann](http://dx.doi.org/10.1017/S2040618500033384), *Proc. Glasgow Math. Assoc.* **3** (1956), 13–17.


