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The enumeration of groups of order \(p^{n}q\) for \(n\leq 5\). (English) Zbl 1403.20035

Having a complete list of isomorphism types of groups of any given order might be seen as the ultimate goal of finite group theory. This is obviously an impossible mission, but over the past 125 years or so advances have been made, starting with results of Hölder and progressing to much more challenging lists, as for example the classification of all groups of order \(p^n\) for \(n\leq 7\) and \(p\) an arbitrary prime.
The paper under review deals with a restricted class of solvable groups.
Let \(p\) and \(q\) be distinct primes and let \(n\leq 5\).The authors determine a function \(\mathcal N_n(p,q)\) that yields the number of isomorphism types of groups of order \(p^nq\).
This function is expressed as the sum of four somewhat simpler functions, describing the number of nilpotent groups in question, the number of non-nilpotent groups with a normal Sylow \(p\)-subgroup, the number of non-nilpotent groups with a normal Sylow \(q\)-subgroup and finally the number of groups in question having no normal Sylow subgroup.
The mathematics involved is complex, for the Lazard correspondence, combinatorial techniques related to various linear groups, generalized polynomials on residue classes modulo various primes are used in combination with linear algebra techniques and various computer algebra algorithms and libraries.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20-04 Software, source code, etc. for problems pertaining to group theory
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