Parafree group
Encyclopedia
In 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 (the intersection of the terms of its lower central series is trivial).
Parafree groups share many properties with free group
s, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the realm of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
and if it is residually nilpotent (the intersection of the terms of its lower central series is trivial).
Parafree groups share many properties with free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s, making it difficult to distinguish between these two types. Gilbert Baumslag
Gilbert Baumslag
Gilbert Baumslag is a Distinguished Professor at the City College of New York, with joint appointments in mathematics, computer science, and electrical engineering. He is director of the Center for Algorithms and Interactive Scientific Software, which grew out of the MAGNUS computational group...
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.