Artin transfer patterns on descendant trees of finite p-groups

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 \(G\mapsto(τ(G),\varkappa(G))\), which consists of families \(τ(G)=(H/H^\prime)_{H\le G}\), 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 \(τ(π(G))\leτ(G)\) and \(\varkappa(π(G))\ge\varkappa(G)\), which are compatible with the parent-descendant relation \(π(G)<G\) of the edges \(G\toπ(G)\) 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 \((τ(G),\varkappa(G))\) 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 \((τ(G),\varkappa(G))\) and explaining the stabilization, resp. polarization, of their components in descendant trees \(\mathcal{T}\) of finite \(p\)-groups.