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Article overview
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Artin transfer patterns on descendant trees of finite p-groups | | Daniel C. Mayer
; | | Date: |
24 Nov 2015 | | Abstract: | Based on a thorough theory of the Artin transfer homomorphism
(T_{G,H}:,G o H/H^prime) from a group (G) to the abelianization
(H/H^prime) of a subgroup (Hle G) of finite index (n=(G:H)), and its
connection with the permutation representation (G o S_n) and the monomial
representation (G o Hwr S_n) of (G), the Artin pattern
(Gmapsto( au(G),varkappa(G))), which consists of families
( au(G)=(H/H^prime)_{Hle G}), resp. (varkappa(G)=(ker(T_{G,H}))_{Hle
G}), of transfer targets, resp. transfer kernels, is defined for the vertices
(Ginmathcal{T}) of any descendant tree (mathcal{T}) of finite
(p)-groups. It is endowed with partial order relations
( au(pi(G))le au(G)) and (varkappa(pi(G))gevarkappa(G)), which are
compatible with the parent-descendant relation (pi(G)<G) of the edges
(G opi(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
(( au(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
(( au(G),varkappa(G))) and explaining the stabilization, resp.
polarization, of their components in descendant trees (mathcal{T}) of finite
(p)-groups. | | Source: | arXiv, 1511.7819 | | Services: | Forum | Review | PDF | Favorites | |
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