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, $D_{p+1}(G)=1$; when $D_p(G)\ne1$ 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 $\mathbb{F}_p G$, and indeed by the quotient $\mathbb{F}_p G/\mathrm{I}^{\,ν+1}$, where $\mathrm{I}$ is the augmentation ideal and $ν$ the length of the Jennings series; for an arbitrary finite $p$-group $G$, the quotient $\mathbb{F}_p G/\mathrm{I}^{\,p}$ determines $G/D_p(G)$; and that every automorphism of $\mathbb{F}_p G$ is the product of an automorphism of $G$ and an automorphism inducing the identity on $\mathrm{I}/\mathrm{I}^2$. 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.