Isologism of groups

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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.