Isologism of groups
Contents
Definition
Definition in terms of a defining set of words
Consider a subvariety of the variety of groups. Denote by a set of words that generate the variety (i.e., a group is in iff all words from are trivial for all tuples of elements from the group).
Consider any group (not necessarily in ). Denote by the marginal subgroup of with respect to the variety and denote by the verbal subgroup of with respect to .
For every word , let be the number of distinct letters used in the word. defines a -ary set map:
By the definitions of marginal and verbal subgroup, the map descends to a set map:
An isologism of groups and with respect to is a pair where is an isomorphism between and , is an isomorphism between and , and for every , we have:
Two groups are termed isologic groups with respect to if there exists an isologism with respect to between them.
Note that the choice of does not matter for this definition, all that matters is that generate the variety .
Definition in terms of homologism
An isologism is an invertible homologism, i.e., a homologism where both the component homomorphisms are isomorphisms.
Facts
Particular cases
Subvariety of the variety of groups | Generating set of words | Marginal subgroup | Marginal factor group | Verbal subgroup | Name for notion of isologism |
---|---|---|---|---|---|
variety containing only the trivial group | trivial subgroup | whole group | whole group | isomorphism of groups | |
variety of abelian groups | commutator | center | inner automorphism group | derived subgroup | isoclinism |
variety of nilpotent groups of class at most | length left-normed commutator | member of upper central series | quotient by this member | 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. |