By F. Rudolf Beyl
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Additional resources for Group Extensions, Representations, and the Schur Multiplicator
Sample text
E) N S)-[R,F] Likewise lation of the associated The given description free is immediate. (e) also. (e) The remaining e' assertion of chosen as a is immediate 33 from naturality. 6 ADDENDA. 5 determines at first. ~O "G ~Q ~ 0 , extensions. The formu- completely, e: R C ~ F---~Q although be the free shall be evaluated. ~ and a, p o s s i b l y 8. (e) ~ab ~ ~ab ~ N ~Uab----*~ab ~ M(Q) natural we confuse to not (e,p,1): surJectlve. (e) as the composite = (R n [F,F])/[R,F] " e . 7 LEMMA. extension.
A) = Here g ~ G ~g ~ t~ acts on >g(ta)l ZG ~U Z (These maps are well-known, ~ HomG(ZG by ~U Z,A) g(a ® t) = (gx) ® t cf. HILTON/STAMMBACH for [I; Prop. x e ZG . 1. ~-I Now B % P ZG e U ~ and Z ; then @U V£,A) The connecting homomorphisms J~B" complexes. of ZG ~ Z by isomorphisms ~ HOmu(VP,VA) This description Wn(e) are respected, is a U-free % V~ ~ VB % VP , ~ HOmu(VP,HOmG(ZG,A)) cochaln V(P) resolution is induced by the standard of chain resp. e. 5) G . 3 DEFINITION. define restriction Res n = Hn(i,B) Hn(U'VB) ~n(e) Z) With G ) T O r n _ I ( B ' , Z G e U 7) in cohomology.
Is quite interesting. 8 as follows. ~Q denotes Q an alternating = x-1[gl,g2 ] reap. the exterior form" abellan Using factor ~ el ~~ ~ given a central billnear form extension by 9: Q ^ Q - A where Q ® Q / ( q ® q I q ~ Q > • vanishes precisely systems, is a homomorphlsm [ , ] @: Q × Q - A a homomorphism Now the "commutator extension. 8) 0 ~ Ext(Q,A) Comparison Q . in WARFIELD [fl], viz. (e) M(Q) ~ Q ^ Q commutator form is Baer's results are readily accessible Q is abelian. group is absolutely abelian.