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Counting \(p\)-groups and Lie algebras using PORC formulae. (English) Zbl 1485.20052

Summary: Counting problems whose solution is PORC were introduced in a famous paper by G. Higman [Proc. Lond. Math. Soc. (3) 10, 566–582 (1960; Zbl 0201.36502)]. We consider two specific counting problems with PORC solutions: the number of isomorphism types of \(d\)-generator class-2 Lie algebras over \(\mathbb{F}_q\) (as a function in \(q)\) and the number of isomorphism types of \(d\)-generator \(p\)-class \(2 p\)-groups (as a function in \(p)\). We prove lower bounds for the degrees of their PORC formulae for all \(d \in \mathbb{N}\) and we determine explicit PORC formulae for \(d \leq 7\).

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
17B05 Structure theory for Lie algebras and superalgebras
20-08 Computational methods for problems pertaining to group theory

Citations:

Zbl 0201.36502

Software:

ClassTwoAlg; OEIS; Magma; GAP
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Full Text: DOI

References:

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