Short Jennings series and the modular isomorphism problem
Abstract
Let p be an odd prime and G a finite p-group whose Jennings series has length at most p, that is, Dp+1(G)=1; when Dp(G)≠1 assume also that G/Dp(G) has nilpotency class at most p−2. We prove that G is determined up to isomorphism by its modular group algebra 𝔽p G, and indeed by the quotient 𝔽p G/I ν+1, where I is the augmentation ideal and ν the length of the Jennings series; for an arbitrary finite p-group G, the quotient 𝔽p G/I p determines G/Dp(G); and that every automorphism of 𝔽p G is the product of an automorphism of G and an automorphism inducing the identity on I/I2. In particular the modular isomorphism problem has a positive answer for all p-groups of exponent p and class less than p, and for all p-groups of class at most p−2 and exponent dividing p2 whose derived subgroup has exponent p. Consequently every counterexample in odd characteristic has class at least p−1, or a derived subgroup of exponent at least p2, or an element of order p3. The proof combines a descent theorem in the truncated group algebra of a free group, valid for every prime, with a relative rigidity theorem for the unipotent automorphisms of truncated group algebras of groups of exponent p, proved through the Lazard correspondence. The bound p is the range in which the logarithm of the group algebra exists.
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BibTeX
@misc{tuzzolo2026short,
title = {{Short Jennings series and the modular isomorphism problem}},
author = {Diego Tuzzolo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40868},
eprint = {2610.40868},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40868},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Short Jennings series and the modular isomorphism problem
AU - Diego Tuzzolo
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40868
UR - https://arxiv.org/abs/2610.40868
AB - Let p be an odd prime and G a finite p-group whose Jennings series has length at most p, that is, D_p+1(G)=1; when D_p(G)≠1 assume also that G/D_p(G) has nilpotency class at most p−2. We prove that G is determined up to isomorphism by its modular group algebra 𝔽_p G, and indeed by the quotient 𝔽_p G/I^ ν+1, where I is the augmentation ideal and ν the length of the Jennings series; for an arbitrary finite p-group G, the quotient 𝔽_p G/I^ p determines G/D_p(G); and that every automorphism of 𝔽_p G is the product of an automorphism of G and an automorphism inducing the identity on I/I^2. In particular the modular isomorphism problem has a positive answer for all p-groups of exponent p and class less than p, and for all p-groups of class at most p−2 and exponent dividing p^2 whose derived subgroup has exponent p. Consequently every counterexample in odd characteristic has class at least p−1, or a derived subgroup of exponent at least p^2, or an element of order p^3. The proof combines a descent theorem in the truncated group algebra of a free group, valid for every prime, with a relative rigidity theorem for the unipotent automorphisms of truncated group algebras of groups of exponent p, proved through the Lazard correspondence. The bound p is the range in which the logarithm of the group algebra exists.
ER -