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Large Wreath Products in Modular Group Rings
Author(s) -
Shalev Aner
Publication year - 1991
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/23.1.46
Subject(s) - mathematics , wreath product , group (periodic table) , unit (ring theory) , abelian group , nilpotent , combinatorics , cyclic group , order (exchange) , bounded function , p group , finite group , group ring , pure mathematics , discrete mathematics , product (mathematics) , mathematical analysis , chemistry , geometry , mathematics education , organic chemistry , finance , economics
Let K be a field of characteristic p > 0, and let G be a locally finite p ‐group. We show that, if the unit group of KG is not nilpotent, then it must involve arbitrarily large wreath products. This may be regarded as an asymptotic generalization of a theorem of D. B. Coleman and D. S. Passman concerning non‐abelian unit groups. The proof relies on the following group‐theoretic result, which extends a classical theorem of B. H. Neumann and J. Wiegold. Let G be any group in which every cyclic subgroup has not more than n conjugates. Then the derived subgroup of G is finite, and its order is bounded above in terms of n .
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