By K. W. Gruenberg
Xiv + 275 pages, selection of casual studies and seminars subject matters contain mounted element unfastened motion, cohomology and homology teams, displays and resolutions, unfastened teams, classical extension thought, finite p-groups, cohomological measurement, extension different types and module idea
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But this implies that each ~lJ = 0 (Proposition l, freeness). Remark. ~" is actually a two-sided ideal. 1-Y when viewed as left F-module. LEMMA ~. Proof. LEMMA 4. ) a right ideal of L F a ~/~ is a right Gfmodule. If a ~ and g = w~ c G, then (~+~A~)g = a w + ~ ' . (1) If ~t Is free ~S_ right ideal of ZF on S~ ~/~" is G-free on S +~J~. (li) if ~L is free as right ideal on S, ~ right ideal on T and is also twq-sided~ then ~ i s is free as free as right ideal on ST. ) Proof. (1) (il) I ] s(ZF) ] s~S s~S is clear.
This is an exceedingly subtle problem and relatively little progress has been made. It was shown recently that D~(G) = G~ (due to G. Higman and D. S. Passi). Of. ,. It is easy to see that the prqblem need only be solved for finite p-groups. We argue thus. If the result holds for all finitely genezated groups ~hen it holds for ~l~ groups. K ~ Gn and Dn(G/K ) = Gn/K , then Dn(G ) = G n. Further, ~f K ~ G , Hence one only needs to consider finitely generated nilpotent groups. Such groups are known to possess a family of normal subgroups intersecting in 1 and whose quotient groups are finite prime power groups.
Dn(G ) = (i+ ~n) ~ G is called the n-th dimension group of G. Clearly, Dl(G ) = G and D2(G ) = G 2 (because (l-w)+ iS an isomorphism of ~/ ~2 onto G/G': cf. 4). clear that Dn(G ) ~ G n (use induction on n). I ~0r o b l e m : Is Dn(G) = G n for all n?. This is an exceedingly subtle problem and relatively little progress has been made. It was shown recently that D~(G) = G~ (due to G. Higman and D. S. Passi). Of. ,. It is easy to see that the prqblem need only be solved for finite p-groups.
Cohomological Topics in Group Theory by K. W. Gruenberg