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 $(\zeta ,\phi )$ where $\zeta$ is an isomorphism between $G/V^{*}(G)$ and $H/V^{*}(H)$, $\phi$ is an isomorphism between $V(G)$ and $V(H)$, and for every $w\in W$, we have:

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

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.