Groups and nilpotent Lie rings whose order is the sixth power of a prime. (English) Zbl 1072.20022
The authors give a complete description (up to isomorphism) of all finite \(p\)-groups of order \(p^6\) (available via computer systems). The result is achieved independently of previous attempts of various authors, which contained errors. On the other hand, the reliability of the results is partially confirmed by some of the previous results. The main difficulty is to distinguish different isomorphism types, while a complete list with possible repeats is comparatively easy to obtain. The new ingredient is finding first a similar complete description of nilpotent Lie rings of order \(p^6\), and then using the Baker-Campbell-Hausdorff formula when the nilpotency class is less than \(p\) (the only case not covered by this scheme – that of \(5\)-groups of class \(5\) of order \(5^6\) – was known earlier). For the history of earlier attempts see the extensive survey in the paper.
Reviewer: Eugenii I. Khukhro (Cardiff)
MSC:
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 17B30 | Solvable, nilpotent (super)algebras |
| 20F40 | Associated Lie structures for groups |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
finite \(p\)-groups; nilpotent Lie rings; Baker-Campbell-Hausdorff formula; power-commutator presentations; groups of order \(p^6\)Online Encyclopedia of Integer Sequences:
Number of groups of order 3^n.Number of groups of order 5^n.
Number of groups of order 7^n.
Number of groups of order prime(n)^5.
Number of groups of order prime(n)^6.
Erroneous version of A271811 (but for odd primes only).
Number of non-abelian groups of order prime(n)^6.
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