Isomorphism types of minimal non-nilpotent groups (Q803277)

The author states, ``The aim of this note is to describe the isomorphism types of minimal non-nilpotent groups.'' Included is the construction of the ``universal'' minimal non-nilpotent groups determined by Gol'fand in 1948 Rédei in 1956. Let P be the (relatively) free group on r generators in the variety defined by the equations \(x^ p=1\) and \([x,y,z]=1\) if p is an odd prime or \(x^ 4=1\) and \([x^ 2,y]=1\) when \(p=2\). Let G be the semidirect product PQ where \(Q=<g>\) is cyclic for order \(q^ m\), q prime, and g induces an automorphism of order q on P. Then minimal non-nilpotent \(G_ 1\) is a homomorph of G. (The primes p and q arise from the well-known properties of \(G_ 1.)\) The author notes: ``A general formula expressing the number of isomorphism types would be tremendously complicated. Here we give it only for the case when k is a prime number.''