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 $(\zeta ,\varphi )$ where $\zeta :G_{1}/V^{*}(G_{1})\to G_{2}/V^{*}(G_{2})$, $\varphi :V(G_{1})\to V(G_{2})$ are homomorphisms, and for every $w\in W$, we have:

$\gamma _{w}(\zeta (x_{1}),\zeta (x_{2}),\dots ,\zeta (x_{n_{w}})=\varphi (\gamma _{w}(x_{1},x_{2},\dots ,x_{n_{w}}))\ \forall \ (x_{1},x_{2},\dots ,x_{n})\in (G_{1}/V^{*}(G_{1}))^{n_{w}}$

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:

${\begin{array}{ccc}(G_{1}/V^{*}(G_{1}))^{n_{w}}&{\stackrel {\zeta \times \zeta }{\to }}&(G_{2}/V^{*}(G_{2}))^{n_{w}}\\\downarrow ^{\gamma _{w,G_{1}}}&&\downarrow ^{\gamma _{w,G_{2}}}\\V(G_{1})&{\stackrel {\varphi }{\to }}&V(G_{2})\\\end{array}}$

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