# Isologism of groups

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## 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.

## 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.