Download Abelian Group Theory. Proc. conf. Oberwolfach, 1981 by R. Göbel, E. Walker PDF

By R. Göbel, E. Walker

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Extra info for Abelian Group Theory. Proc. conf. Oberwolfach, 1981

Example text

G I. ~' N I C(&) with A is a normal subgroup of ® with A. 6 we have ~(h) with It is left as an exercise to show that = iV, A A A. 8, ~ I = ~JA, It is an extension of Exercise. 10 PROPOSITION. ® ~ 6(A) exactly once. Then the followin$ statements are equivalent. is the direct product o f PROOF. A be a semidirect product o f ~:H - 6(A). G PROOF. G = HA, H n A = i ([i], § 50). 9 PROPOSITION. H A. intersects each coset of by by the h o m o m o r p h i s m with In terms of (group) extensions, G A extensions see R~dei H A.

S x¢ to it. in of G is the G, let HK = [hk lh E H, k E K]. PROPOSITION. 1 Further, A a normal subsrou~ is the centralizer PROOF. o_f~ ~ , ~ onto subgroup 6(£) o_~f ~ PROOF. 21, M and thus that ~ [ for any M and is a normal M ~ A = im maps and maps ~ into of ~ subgroup I ~ ~ C(a-)} onto of ~, M. I i__~s = C(M). ~. 13, = ~(w,a)(~,b) M Hence ¢ M maps is a normal ~ onto 6(£). = (ab) -i c (ab) = c¢(w~,ab) and hence ~ is a homomorphism. is s normal subgroup of m I $(~). we have Hence 7 E g , we have to show that onto = b -I (a-lca)b (a-lca){(w,b) ~(w,a)~(~,b) ((~,a) E >~), is a homomorphism I/M ~ $(£).

3 COROLLARY. are equivalent: The following conditions on a semilinear isomorphism i) ~ E $(&), ii) ~(~,a) E ~(£), iii) m a (~,a) is linear for some rE&. PROOF. Exercise. 4 COROLLARY. The identity m a p p i n g i~s ~h___eeonly a u t o m o r ~ h i s m of leaves fixed every element of PROOF. Let automorphism ~ (~,a) Hence (w,a) E M S PROOF. (~,a) and £(g,V) Sl S an___~d S ~ ~ and ~(g',V~), so t h a t ~(w,a)~(i',a ') these are extensions of b of Now ~2,U(&,V), If then PROOF. 4, and s i n c e ~ = ~.

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