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An explicit construction of a limit variety of groups

  • Zidong Gao
  • Yilin Zhou
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Abstract

We construct an explicit countable group G*=N(R)rtimes A whose generated variety is a limit variety of groups: it has no finite identity basis, whereas every proper subvariety is finitely based. Here A is relatively free in the variety of groups of nilpotency class at most two and exponent dividing four, and R is an explicitly defined quotient of the group algebra 𝔽2[A]. The proof establishes an ascending chain condition for coefficient spaces indexed by six ordered commutator entries and reduces every proper subvariety to a hereditarily finitely based approximation. Non-finite basability is obtained by transferring independent identities from the constructions of Gupta and Krasilnikov. The resulting variety is contained in B4B2. This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026explicit,
  title = {{An explicit construction of a limit variety of groups}},
  author = {Zidong Gao and Yilin Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17086},
  eprint = {2610.17086},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17086},
}
RIS
TY  - UNPB
TI  - An explicit construction of a limit variety of groups
AU  - Zidong Gao
AU  - Yilin Zhou
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17086
UR  - https://arxiv.org/abs/2610.17086
AB  - We construct an explicit countable group G_*=N(R)rtimes A whose generated variety is a limit variety of groups: it has no finite identity basis, whereas every proper subvariety is finitely based. Here A is relatively free in the variety of groups of nilpotency class at most two and exponent dividing four, and R is an explicitly defined quotient of the group algebra 𝔽_2[A]. The proof establishes an ascending chain condition for coefficient spaces indexed by six ordered commutator entries and reduces every proper subvariety to a hereditarily finitely based approximation. Non-finite basability is obtained by transferring independent identities from the constructions of Gupta and Krasilnikov. The resulting variety is contained in B_4B_2. This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.
ER  -