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A New Characterisation of m ‐Tame Groups Over Finitely Generated Abelian Groups
Author(s) -
Kochloukova Dessislava H.
Publication year - 1999
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008194
Subject(s) - mathematics , abelian group , nilpotent group , pure mathematics , conjecture , quotient group , quotient , nilpotent , prime (order theory) , group (periodic table) , finitely generated abelian group , commutator subgroup , combinatorics , cyclic group , discrete mathematics , normal subgroup , physics , quantum mechanics
It is shown that, if Q is a finitely generated abelian group, a finitely generated Q ‐group A is m ‐tame if and only if the m th tensor power of the augmentation ideal of Z A is finitely generated over A m ⋊ Q , where Q acts diagonally on both A m and the tensor power. It is proved that quotients of metabelian groups of type FP 3 are again of type FP 3 , and a necessary condition is found for a split extension of abelian‐by‐(nilpotent of class two) groups to be of type FP 2 . A conjecture is formulated that generalises the FP m ‐Conjecture for metabelian groups, and it is shown that one of the implications holds in the prime characteristic case.