The Groups and Nilpotent Lie Rings of Order <i>p</i> ⁸ with Maximal Class (Q6564874)

Let \(p\) be a prime number an let \(f(p,n)\) be the number of isomorphism classes of groups of order \(p^{n}\). In 1960, \textit{G. Higman} wrote two important papers [Proc. Lond. Math. Soc., III. Ser. 10, 24--30 (1960; Zbl 0093.02603); Proc. Lond. Math. Soc., III. Ser. 10, 566--582 (1960; Zbl 0201.36502)], in which he conjectured that \(f(p, n)\) is a PORC (\(\mathsf{P}\)olynomial \(\mathsf{O}\)n \(\mathsf{R}\)esidue \(\mathsf{C}\)lasses) function of \(p\) for each positive integer \(n\). A function on primes is PORC if and only if it is PORC on all but finitely many primes, because the finitely many primes can be incorporated by multiplying them with the modulus of the residue classes. Thus, the conjecture can be reformulated as stating that \(f(p,n)\) is a PORC function of \(p\) for all sufficiently large \(p\).\N\NHigman's PORC conjecture was proved for \(n \leq 7\) (for the case \(n=7\), see the paper of \textit{E. A. O'Brien} and the second author [J. Algebra 292, No. 1, 243--258 (2005; Zbl 1108.20016)]). The reviewer points out that evidence was found in [\textit{M. du Sautoy} and the second author, J. Algebra 361, 287--312 (2012; Zbl 1267.20025)] that the PORC conjecture is not valid for \(n \geq 10\).\N\NLet \(m(p,n)\) be the number of groups of order \(p^{8}\) with maximal class. In the paper under review, the authors prove that \(m(p,n)\) is a PORC function for \(p \geq 11\). To do this, they first prove Theorem 1.1: For \(p \geq 5\) the number of nilpotent Lie rings of order \(p^{8}\) which have maximal class is\N\[\N4p^{3}+7p^{2}+9p+6+(6p+11)g_{3}+ 4g_{5}+(p+2)g_{7}+(p+3)g_{8}+2g_{9}+g_{12},\N\]\Nwhere \(g_{i}=\gcd(p-1,i)\). Therefore, they apply the Lazard correspondence [\textit{M. Lazard}, Ann. Sci. Éc. Norm. Supér., III. Sér. 71, 101--190 (1954; Zbl 0055.25103)] between \(p\)-groups and nilpotent Lie rings.