Isologism 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:

${\beta }_w:G^{n_w}\to G$

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

${\gamma }_w:(G/V^{*}(G){)}^{n_w}\to V(G)$

An isologism of groups $G$ and $H$ with respect to $\mathcal{V}$ is a pair $(\zeta ,\varphi )$ where $\zeta$ is an isomorphism between $G/V^{*}(G)$ and $H/V^{*}(H)$, $\varphi$ is an isomorphism between $V(G)$ and $V(H)$, and for every $w\in W$, we have:

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

Two groups are termed isologic groups with respect to $\mathcal{V}$ if there exists an isologism with respect to $\mathcal{V}$ between them.

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

Definition in terms of homologism

An isologism is an invertible homologism, i.e., a homologism where both the component homomorphisms are isomorphisms.

Facts

• Isologism with respect to variety is isologism with respect to any bigger variety

Particular cases

Subvariety of the variety of groups	Generating set of words	Marginal subgroup $V^{*}(G)$	Marginal factor group $G/V^{*}(G)$	Verbal subgroup $V(G)$	Name for notion of isologism
variety containing only the trivial group	$x$	trivial subgroup	whole group	whole group	isomorphism of groups
variety of abelian groups	commutator $[x_1,x_2]$	center	inner automorphism group	derived subgroup	isoclinism
variety of nilpotent groups of class at most $c$	length $c+1$ left-normed commutator	$c^{th}$ member of upper central series	quotient by this member	$(c+1{)}^{th}$ member of lower central series	fixed-class isoclinism
variety of all groups	empty word, i.e., a word that is always the identity element	whole group	trivial group	trivial group	no name, all groups are isologic.