Artin transfer patterns on descendant trees of finite p-groups

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.

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