Year: 2026
Type: preprint
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 \(\mathbb F_2[A]\). 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 \(\mathfrak B_4\mathfrak B_2\). This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.
Source: arXiv (Cornell University)
Authors: Gao, Zidong, Zhou, Yilin
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221799919