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From Mislin Genus to Strong Genus
Author(s) -
Cassidy Charles,
Scevenels Dirk
Publication year - 1999
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/s0024609398005190
Subject(s) - mathematics , genus , bounded function , combinatorics , counterexample , cardinality (data modeling) , finitely generated group , discrete mathematics , pure mathematics , finitely generated abelian group , mathematical analysis , botany , computer science , data mining , biology
If two finitely generated, torsion‐free, nilpotent groups of class two satisfy the two‐arrow property (that is, they embed into each other with finite, relatively prime indices), then they necessarily belong to the same Mislin genus (that is, they have isomorphic localizations at every prime). Here we show that the other implication is false in general. We even provide counterexamples in the case where both groups have isomorphic localizations at every finite set of primes of bounded cardinality. The latter equivalence relation leads us to introduce the notion of n ‐genus for every positive integer n , which we show to be meaningful in various contexts. In particular, the two‐arrow property is related to the n ‐genus in the context of topological spaces. 1991 Mathematics Subject Classification 20F18, 55P60.