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arXiv · 2609.06166

The entropy of coclass one and of SmallGroup(729,45)

Helga Boyer von Berghof
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Abstract

Investigating the descendant tree T(Delta) of 3-groups G of maximal nilpotency-class, with root Delta=SmallGroup(27,3), we show for the first time that an entropy-value H(P_r)=sum P_r(G)|log(P_r(G))| can also be determined for the real probability-distribution P_r(G) of all terminal vertices outside of the mainline on this infinite digraph. These vertices arise as automorphism groups Gal(F(3,infinity,K)/K) of maximal unramified pro-3-extensions F(3,infinity,K) of real-quadratic number fields K=Q(d^1/2), d>0. For the groups G of another descendant tree T(N), with root N=SmallGroup(729,45) and unbounded coclass, which has been studied by L. Bartholdi and M. R. Bush, we compare the entropy H(P_r) of the real measure P_r(G) with the entropy H(P_i) of the imaginary measure P_i(G).

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Helga Boyer von Berghof. 2026-09-05. The entropy of coclass one and of SmallGroup(729,45). https://arxiv.org/abs/2609.06166

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