Reduced free group
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Contents
Definition
No. | Shorthand | A group is termed a reduced free group if ... | A group ![]() |
---|---|---|---|
1 | Quotient of free | it is isomorphic to the quotient of a free group by a verbal subgroup | there is a free group ![]() ![]() ![]() ![]() |
2 | Free in some subvariety | it is a free algebra in some subvariety of the variety of groups. | there is a subvariety ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | Free in own subvariety | it is a free algebra in the subvariety of the variety of groups generated by itself. | ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Free group | free in the variety of groups | |FULL LIST, MORE INFO | ||
Free abelian group | free in the variety of abelian groups | |FULL LIST, MORE INFO | ||
Elementary abelian group | trivial or abelian of prime exponent | |FULL LIST, MORE INFO | ||
Burnside group | free in the variety of groups of exponent dividing ![]() ![]() |
|FULL LIST, MORE INFO | ||
Finite homocyclic group | finite direct power of a finite cyclic group | |FULL LIST, MORE INFO | ||
Free class two group | free in the variety of groups of nilpotency class two | |FULL LIST, MORE INFO | ||
Free metabelian group | free in the variety of metabelian groups | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Group in which every fully invariant subgroup is verbal | every fully invariant subgroup is a verbal subgroup | fully invariant implies verbal in reduced free | simple groups give counterexamples | |FULL LIST, MORE INFO |