Artin transfer patterns on descendant trees of finite p-groups

TL;DR

This paper develops a theoretical framework for analyzing Artin transfer patterns on descendant trees of finite p-groups, enabling more efficient identification and classification of such groups.

Contribution

It introduces a new partial order structure on Artin patterns, facilitating termination criteria for p-group generation algorithms and advancing the understanding of transfer behaviors.

Findings

Defined the Artin pattern for vertices in descendant trees

Established partial order relations compatible with tree structure

Provided criteria for terminating p-group generation algorithms

Abstract

Based on a thorough theory of the Artin transfer homomorphism \(T_{G,H}:\,G\to H/H^\prime\) from a group \(G\) to the abelianization \(H/H^\prime\) of a subgroup \(H\le G\) of finite index \(n=(G:H)\), and its connection with the permutation representation \(G\to S_n\) and the monomial representation \(G\to H\wr S_n\) of \(G\), the Artin pattern \(G\mapsto(\tau(G),\varkappa(G))\), which consists of families \(\tau(G)=(H/H^\prime)_{H\le G}\), resp. \(\varkappa(G)=(\ker(T_{G,H}))_{H\le G}\), of transfer targets, resp. transfer kernels, is defined for the vertices \(G\in\mathcal{T}\) of any descendant tree \(\mathcal{T}\) of finite \(p\)-groups. It is endowed with partial order relations \(\tau(\pi(G))\le\tau(G)\) and \(\varkappa(\pi(G))\ge\varkappa(G)\), which are compatible with the parent-descendant relation \(\pi(G)<G\) of the edges \(G\to\pi(G)\) of the tree \(\mathcal{T}\). The partial order enables termination criteria for the \(p\)-group generation algorithm which can be used for searching and identifying a finite \(p\)-group \(G\), whose Artin pattern \((\tau(G),\varkappa(G))\) is known completely or at least partially, by constructing the descendant tree with the abelianization \(G/G^\prime\) of \(G\) as its root. An appendix summarizes details concerning induced homomorphisms between quotient groups, which play a crucial role in establishing the natural partial order on Artin patterns \((\tau(G),\varkappa(G))\) and explaining the stabilization, resp. polarization, of their components in descendant trees \(\mathcal{T}\) of finite \(p\)-groups.