The groups and nilpotent Lie rings of order \(p^8\) with maximal class. (English) Zbl 07873971

Lee, Seungjai; Vaughan-Lee, Michael

Exp. Math. 33, No. 2, 247-253 (2024).

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, 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 \(p\).
Higman’s PORC conjecture was proved for \(n \leq 7\) (for the case \(n=7\), see the paper of 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 [M. du Sautoy and the second author, J. Algebra 361, 287–312 (2012; Zbl 1267.20025)] that the PORC conjecture is not valid for \(n \geq 10\).
Let \(m(p,n)\) 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 \[ 4p^{3}+7p^{2}+9p+6+(6p+11)g_{3}+ 4g_{5}+(p+2)g_{7}+(p+3)g_{8}+2g_{9}+g_{12}, \] where \(g_{i}=\gcd(p-1,i)\). Therefore, they apply the Lazard correspondence [M. Lazard, Ann. Sci. Éc. Norm. Supér., III. Sér. 71, 101–190 (1954; Zbl 0055.25103)] between \(p\)-groups and nilpotent Lie rings.

Reviewer: Enrico Jabara (Venezia)

MSC:

20D15	Finite nilpotent groups, \(p\)-groups
17B30	Solvable, nilpotent (super)algebras

Keywords:

\(p\)-group; maximal class; Lie \(p\)-ring; Higman’s PORC conjecture

Citations:

Zbl 0093.02603; Zbl 0201.36502; Zbl 1108.20016; Zbl 1267.20025; Zbl 0055.25103

Software:

GAP; LiePRing; Magma; LieRing

References:

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