Homologism of groups

Definition

Definition in terms of a defining set of words

Consider a subvariety $\mathcal{V}$ of the variety of groups. Denote by $W$ a set of words that generate the variety $\mathcal{V}$ (i.e., a group is in $\mathcal{V}$ iff all words from $W$ are trivial for all tuples of elements from the group).

Consider any group $G$ (not necessarily in $\mathcal{V}$). Denote by $V^{*}(G)$ the marginal subgroup of $G$ with respect to the variety $\mathcal{V}$ and denote by $V(G)$ the verbal subgroup of $G$ with respect to $\mathcal{V}$.

For every word $w\in W$, let $n_w$ be the number of distinct letters used in the word. $w$ defines a $n_w$-ary set map:

${\beta }_{w,G}:G^{n_w}\to G$

By the definitions of marginal and verbal subgroup, the map descends to a set map:

${\gamma }_{w,G}:(G/V^{*}(G){)}^{n_w}\to V(G)$

A homologism of groups $G_1$ and $G_2$ with respect to $\mathcal{V}$ is a pair $(\zeta ,\phi )$ where $\zeta :G_1/V^{*}(G_1)\to G_2/V^{*}(G_2)$, $\phi :V(G_1)\to V(G_2)$ are homomorphisms, and for every $w\in W$, we have:

${\gamma }_w(\zeta (x_1),\zeta (x_2),\dots ,\zeta (x_{n_w})=\phi ({\gamma }_w(x_1,x_2,\dots ,x_{n_w}))\mathrm{\forall }(x_1,x_2,\dots ,x_n)\in (G_1/V^{*}(G_1){)}^{n_w}$

Note that the choice of $W$ does not matter for this definition, all that matters is that $W$ generate the variety $\mathcal{V}$.

In other words, for every $w\in W$, the following diagram commutes:

$\begin{array}{ccc}(G_1/V^{*}(G_1){)}^{n_w}& & (G_2/V^{*}(G_2){)}^{n_w}\\ {\downarrow }^{{\gamma }_{w,G_1}}& & {\downarrow }^{{\gamma }_{w,G_2}}\\ V(G_1)& & V(G_2)\\ \end{array}$

Note that the choice of the defining set of words does not matter, i.e., if $W_1$ and $W_2$ are different sets of words that generate the same variety $\mathcal{V}$, the condition of being a homologism with respect to $W_1$ coincides with the condition of being a homologism with respect to $W_2$.

Related notions

Term	Meaning
category of groups with homologisms	this is a category whose objects are group and where the morphisms are homologisms of groups.
isologism of groups	this is a homologism where both the component homomorphisms are isomorphisms.

Particular cases

Variety	Generating word or set of words	Corresponding notion of homologism
variety of abelian groups	$[x_1,x_2]$ -- the commutator word	homoclinism of groups
variety of groups of nilpotency class at most $n$	$[[\dots [[x_1,x_2],x_3],\dots ,x_c],x_{c+1}]$ -- the left-normed iterated commutator word	n-homoclinism of groups