The Nil K-groups of finite groups
Abstract
We show that for every finite group G and every integer n, the Nil group NKn(ℤ[G]) of the integral group ring of G has finite exponent, and give a bound depending on n and the order |G|. This bound is explicit when every prime divisor of |G| is at least 3+n/2. More generally, our finite exponent result applies to NKn(R[G]) whenever R is a regular, torsion-free, Noetherian commutative ring such that R/p is regular for every prime p that divides |G|. In the Appendix, M. Morrow provides an alternative approach and also shows that the Nil group NKn(X) of an excellent Noetherian scheme X, with X[1/p] regular, is annihilated by a finite power of p, provided X admits a suitable resolution of singularities. By extending his argument to certain noncommutative rings, we also show that, for R as above and α any automorphism of G, the Farrell Nil groups NKn(R[G],α) have finite exponent. As a consequence, for every virtually cyclic group Γ, the group Kn(ℤ[Γ]) is a direct sum of a finitely generated abelian group and an infinite countable direct sum of copies of a finite abelian group.
Related
- Finite abelian groups are Schur multipliers of finite solvable groups
- Finite abelian subgroups of algebraic groups
- Representation zeta functions of wreath products with finite groups
- Element Orders and Character Codegrees of Finite Groups
- Finite groups with large sum of element orders compared to cyclic groups
BibTeX
@misc{chinburg2026kgroups,
title = {{The Nil K-groups of finite groups}},
author = {Ted Chinburg and Matthew Morrow and Georgios Pappas and Martin J. Taylor},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13908},
eprint = {2609.13908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13908},
}
RIS
TY - UNPB
TI - The Nil K-groups of finite groups
AU - Ted Chinburg
AU - Matthew Morrow
AU - Georgios Pappas
AU - Martin J. Taylor
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13908
UR - https://arxiv.org/abs/2609.13908
AB - We show that for every finite group G and every integer n, the Nil group NK_n(ℤ[G]) of the integral group ring of G has finite exponent, and give a bound depending on n and the order |G|. This bound is explicit when every prime divisor of |G| is at least 3+n/2. More generally, our finite exponent result applies to NK_n(R[G]) whenever R is a regular, torsion-free, Noetherian commutative ring such that R/p is regular for every prime p that divides |G|. In the Appendix, M. Morrow provides an alternative approach and also shows that the Nil group NK_n(X) of an excellent Noetherian scheme X, with X[1/p] regular, is annihilated by a finite power of p, provided X admits a suitable resolution of singularities. By extending his argument to certain noncommutative rings, we also show that, for R as above and α any automorphism of G, the Farrell Nil groups NK_n(R[G],α) have finite exponent. As a consequence, for every virtually cyclic group Γ, the group K_n(ℤ[Γ]) is a direct sum of a finitely generated abelian group and an infinite countable direct sum of copies of a finite abelian group.
ER -