In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the realm of
group theory, a
group is said to be parafree if its quotients by the terms of its
lower central series are the same as those of a
free group and if it is
residually nilpotent In the mathematics, mathematical field of group theory, a group is residually ''X'' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''".
Formally, a group ''G'' is residually ''X'' if for every non-triv ...
(the intersection of the terms of its lower central series is trivial).
Parafree groups share many properties with
free groups, making it difficult to distinguish between these two types.
Gilbert Baumslag was led to the study of parafree groups in attempts to resolve the conjecture that a group of cohomological dimension one is free. One of his fundamental results is that there exist parafree groups that are not free. With Urs Stammbach, he proved there exists a non-free parafree group with every countable subgroup being free.
References
*Baumslag, Gilbert, ''Groups with the same lower central sequence as a relatively free group. I. The groups.''
Trans. Amer. Math. Soc.
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...
129 1967 308--321.
*Baumslag, Gilbert; Stammbach, Urs, ''A non-free parafree group all of whose countable subgroups are free.'' Math. Z. 148 (1976), no. 1, 63--6
External links
Parafree one-relator groups
Properties of groups
{{group-theory-stub