P-localization of some classes of groups

Publicacions Ma emá iques, Vol 37 (1993), 19-44 . Abs ac P-LOCALIZATION OF SOME CLASSES OF GROUPS AUGUSTO REYNOL FILHO The aim o he p esen pape is o s udy he heo y o P- Localiza ion o a g oup in a ca ego y C such ha i con ains he ca ego y o he nilpo en g oups as a ull sub-ca ego y . In he sec- ond sec ion we p esen a numbe o esul s on P-localiza ion o a g oup G, which is he semi-di ec p oduc o an abelian g oup A wi h a g oup X, in he ca ego y G o all g oups . I ums ou ha he P-localized (Gp) is comple ely desc ibed by he P-localized Xp o X, A and he ac ion w o X on A . In he hi d sec ion, we p esen he cons uc ion o he heo y o P-localiza ion in he ca ego y o all g oups which a e ex ensions o nilpo en g oups by ini e abelian g oups . Ou p oo ollows a he closely he one p e- sen ed in [2, chap e II, and is based on he classical in e p e a ion o he second cohomology g oup o a g oup . In oduc ion Since Sulli an i s poin ed ou he a ailabili y and applicabili y o localiza ion me hods in homo opy heo y, he e has been conside able wo k done on u he de elopmen s and e inemen s o he me hod and on he s udy o new a eas o applica ion . In [2] P . Hil on, G . Mislin and J . Roi be g cons uc ed he heo y o P-localiza ion o nilpo en g oups, whe e P is a se o p imes . Some ime la e , P . Ribenboin in [3] showed ha i was possible o localize any g oup . (The e is ano he ap- p oach conce ning P-localiza ion in g oup heo y de eloped by Bous ield in Topology 14 (1975) 133-150, and Mem . Ame . Ma h . Soc . 1 0 (1977) no . 186, bu , in his wo k, we jus use he concep s p esen ed in [2], [3] and [4]) . The cons uc ion p esen ed in [3], howe e , seems o be qui e abs ac and his led us o y o ob ain a mo e explici cons uc ion o he P-localiza ion o a g oup G in he ca ego y o all g oups . We we e success ul when G is a semi-di ec p oduc o a ini e abelian g oup A by 2 0  A . REYNOL FILHO some o he g oup X . In addi ion, we managed o cons uc heo y o P- localiza ion o a g oup in he ca ego y C o g oups which a e ex ensions o nilpo en g oups by ini e abelian g oups . The ques ion conce ning semi-di ec p oduc is aken up in Sec ion 2and he main esul s a e 2 .1, 2 .5 and 2 .10 which could be s a ed as ollows . Le P' be he complemen a y o P in he se o all p imes . Theo em (2 .1) Le N -±-> G -'» X be an exac sequence o g oups, whe e N is a p-g oup and p E P' . Then, e = eooEP-localizes G, p o ided ha X  Xp P-localizes X . In his con ex , Theo em 2 .5 says he ollowing : Le X  w ) Au (A) be an ac ion, whe e A is a ini e abelian p-g oup and p E P . Le P l = {q E P' : q ~~w(X) ~} and conside Pi he mul iplica i e se gene a ed by P, . Se H he sub-g oup o X gene a ed by all x E X such ha he o de o w(x) belongs o P,' . Le wH be he es ic ion o w o H and F = l'F7, whe e is he smalles posi i e in ege such ha F'H = I"Hl (He e Fi has he o dina y meaning and i s de ini ion may be ound in [2]) . The e is an ac ion X  Au (A/ ) induced by w, which can be ac o ed as X  w )  Au (A/ ) Le G = Al ~,X and G' = A/ 1 W ,Xp Then, e : (a,, x) E G ---> (a+ F, eo(X)) E G' P-localizes G . Finally, Theo em 2 .10 analyzes he si ua ion in which A is a ini e abelian g oup . Le X ~ w _-> Au (A) be an ac ion, whe e A is a ini e abelian g oup . Le A l . . . . , A be he p-p ima y componen e o A and wi : X ---> Au (Ai) be he ac ions induced by w . - Le G = A ] c~X,Gi = A 2 ]  ,X and ake G ~ E Xp o be he pull- (EJp back o he a ows (Gi) p  -*  Xp . (No ice ha (Gi) p a e gi en by ei he 2 .1 o 2 .5 . Then, we claim ha he na u al homomo phism G G P-localizes G . We de o e Sec ion 3 o p esen ou esul e conce ning he cons uc ion o he heo y o P-localiza ion o a g oup in he ca ego y C o g oups P-LOCALIZATION OF SOME CLASSES OFGROUPS  2 1 which a e ex ensions o nilpo en g oups by ini e abelian g oups . The ma e could be desc ibed as ollows : Gi en G El C 1, he e exis s a unique ini e abelian sub-g oup U o G such ha G/U is nilpo en and I'2 =U whe e W is he ac ion a - ached o he ex ension : U N G -» G/U . Fu he mo e, he e ex- is s a unique g oup U and an epimo phism p : U -» U and a unique bP E H Z ((G1U)P ; i7) . (bp : U - GP - (Gl u)P) a ached o 1 yielding commu a i i y in he diag am U G Gw I PU  e  l eo jP :  _  ,--,  G P  -~  (Gw)P Unde such condi ions (3 .12) s a es ha G  > G P is a unc o and e is a na u al ans o ma ion o unc o s . In addi ion, (3 .13) also s a es ha eP-localizes G . In Sec ion 1 we in oduce some basic esul s needed in he ollowing sec ion . We belie e, ne e heless, ha Theo em (1 .21) is impo an on i s own acco d ; i s a es ha i X -~ Au (A) is a comu a i e diag am whe e X ~- e-'> X P P-localizes X and A is a P- local ini e abelian g oup, hen eo* : H,1 P (X P ; A)  H, 1 (X ; A) is an isomo phism . This wo k is he main pa o he au ho 's Ph .D . hesis done unde he guidance o P o esso Pe e John Hil on . The au ho is e y much indeb ed o P o esso Dacibe g Lima Gonjal es, a whose sugges ion his wo k was de eloped . 1 . P elimina ies In his sec ion we in oduce some gene al esul s on P-local g oups, ac o ing o ac ions and some p oposi ions conce ning g oup cohomology . We s a by ixing he no a ions P' = {n E N* : p 1 n => p E P} = mul iplica i e se gene a ed by P ; P' is he complemen a y o P in he se o all p imes . We also ecall ha G is said o be a P-local g oup  ((dn E P'm)x E G ~--> x' E G is bijec i e) . 22  A . REYNOL FILHO Mo eo e , G  e ) GP P-localizes G E l C 1 in he ca ego y C 4=>(GP E IC 1, GP is P-local and b'H E I C 1, H P-local, (V E HOm(G, H»(3! p E Hom(GP, H)) which yields commu a i i y in he diag am G --~, H el  l p  ) . G P P oposi ion 1 .1 . Le E,, ... , E , K be P-local g oups, whe e P is a se o p imes . Le el E Hom(Ej, K), i = 1, . . . , . In hese condi ions, i E E Hom(E, K) is he pull-back o he amily (Ej)I<j< , hen E is P-local . P oo . S aigh o wa d . P oposi ion 1 .2 . Le 0 E Hom(Y F), whe e Y is a P-local g oup and F is a ini e g oup . Then, (dy E Y) we ha e o(O(y)) =n E Px ; (o(O(y)) = o de o O(y)) . P oo .. Le yE Y and suppose ha 3q E P' wi h q 1 o(O(Y)) . Then we may conside z = y k , whe e o(o(y)) = q .k . I ollows ha o(o(z)) = q . The ac ha Y is P-local and q E P' enables us o s a e ha d > 0, 3z E Y such ha z4 = z . Thus, O(z )g = O(z) ~ 1 and 4'(z )g +' - O(z) 4 = 1, SO o(O(z )) = q +l, y > 0 . In pa icula {O(z ) E F : > 0} is in ini e . Howe e his is impossible, since F is ini e . Co olla y 1 .3 .  Unde he condi ions o he p e ious p oposi ion, we ha e 1 O(Y) ¡ E Px (ie, O(Y) is a P- o sion ( ini e) sub-g oup o F) . Rema k 1 .4 . Acco ding o p oposi ion (7 .1) in [4] we ha e ha a ini e g oup F is P-local  F is a P- o sion g oup . The ull subca ego y o he ca ego y 9 o all g oups consis ing o all nilpo en g oups is deno ed by l . P oposi ion 1 .5 . Le X -- w -> Au (N) be an ac ion, whe e X and N a e g oups wi h Au (N) ini e . Then 3 !wp such ha he diag am X Au (N) eo i XP is commu a i e  w(X) is a P- o sion sub-g oup o Au (N) . P oo . I ollows di ec ly om Co olla y 1 .3 and Rema k 1 .4 . P-LOCALIZATION OF SOME CLASSES OF GROUPS  2 3 Now we ake A Ñ G -'!» X an exac sequence o g oups, whe e A is abelian . Le X -- w -> Au (A) be he ac ion de ined by p(w((x) .a) = g.p(a) .g-1 (whe e E (g) = x) . Fix a collec ion o p imes P, n E N and x E X, and de ine : en(x) = lA + w(x) + . . . +w(xn -1 ) E End(A) . Fo his endomo phism we ha e : Lemma 1.6 . (,u(a) . g) n = p(0(E(g)) .a)gn ; b'g E G, da E A, dn E N . P ooE I is easy by induc ion on n . P oposi ion 1 .7 . Le P be a se o p imes and le A ~ G ~ X be an exac sequence o g oups, whe e A is abelian and w is he ac ion a ached o he ex ension . Fix he condi ions : (i) G is P-local ; (ii) X is P-local ; (iii) 9 n (x) E Au (A),Vx E X,`dn E P" . Then, i wo o (i) ; (ii) ; (iii) ; hold so does he hi d . P oo .. (ii) + (iii) ==> (i) . Fix n E P" . Le g, h E G and suppose ha gn = hn . Then, E(g)n = E(h)n => E(g) = E(h) since X is P-local . So, g = p(a) .h and hn = gn = (p(a) .g)n = p(0 .(E(h)) .a)hn (Lemma 1 .6) . Hence B n (E(h)) .a = 0 . So a=0 andg=h . Likewise, le g E GAx E X such ha E(g) = xn (XP-local) . The e- o e, e(g) = xn = E(hn) . The e o e g = u(a) .hn . Take b E A such ha a = Bn(e(h)) .b . Thus g = F¿(B,(E(h)) .b) .hn = (p(b) .h)n due o 1 .6 . So g E G  > gn E G is bijec i e . (The o he implica ions a e simila ) . P oposi ion 1 .8 . Le P be a se o p imes and le A Ñ G  X be an exac sequence o g oups, whe e A is ini e abelian .  Then, Bn(x) E Au (A), dx E X, dn E P", p o ided ha ei he G is P-local o A and X a e P-local . P oo . (I) G is P-local . Fix n E P` and x E X . Suppose ha Bn (x) .a = 0 . Le g E G such ha E(g) = x . Then (p(a) .g)n = p(Bn(x) .a)g n = gn . So p(a) .g = g and a = 0 . Thus en(x) E Au (A) since A is ini e . (II) A and X a e P-local . Fix n E P` and x E X, and suppose 0,(x) .a = 0 . Thus, (w(xn) - lA) .a = (w(x) - 1A)00 n (x) .a = 0 he e o e w(x') .a = a . On he o he 2 4  A . REYNOL FILHO hand, o(w(x)) = m E P' (P op . 1 .2) . The e o e w(x)m .a = a . As gcd(m, n) = 1, i ollows ha w(x) .a = a, whence 0 = B n (x) .a = n .a . Thus a = 0 since A is P-local . So 8,,,(x) E Au (A) . a Co olla y 1 .9 . In he condi ions o he p oposi ion aboye (1 .8), G is P-local <-==~ A and X a e P-local . E - "I Nex we conside a spli ex ension N >4 G  X, whe e N is a i- 0 ni e g oup . Le X --- w -> Au (N) be he ac ion gi en by p(w(x) .a) = u(x) .p(a) .o,(x)-1 and ake Bn(x) : N --- N de ined by B n (x) .a = 1A(a) .(w(x) .a) . . . (w(x)n -1 . a), x EX ; n EN* : In his sligh ly di e en con ex we now desc ibe p ope ies which a e qui e simila o P op . 1 .6, 1 .7, 1 .8 and 1 .9 . Lemma 1 .10 . (p(a) .(x))n = P( n (x) .a) o . (X)n ; dx E X ; dn E N* ; `da E A . P oo . See P op . 1 .6 . P oposi ion 1 .11 . Le N Ñ G <--< X be a spli sho exac sequence o g oups and le w be he ac ion de ined by he spli ing o .  Fix he s a emen s : (i) G is P-local ; (ii) X is P-local ; (iii) en(x) is a bijec ion, bx E X ; `dn E P'x . Then, i wo o (i) ; (ü) ; (iii) ; hold, so does he hi d . P oo . See P op . 1 .7 . E P oposi ion 1 .12 . Le N >-~ G E--< X be a spli sho exac sequence 0 o g oups, whe e N is ini e . Then Bn(x) is a bijec ion, dn E P", dx E X p o ided ha ei he G is P-local o N and X a e P-local . P oo . I G is P-local, hen he p oo ollows as (I) P op . 1 .8 . So le 's suppose ha N and X a e P-local . Le w(X) ~- i > Au (N) and G= N 1 Zw(X) . w(X) is a P- o sion g oup (Co . 1 .3) and N is a P- o sion g oup (Rema k 1 .4) . Thus G is a P- o sion g oup . So G is P-local (Rema k 1 .4) . E Then aking he sequence N ~ G <Z w(X) and in oking he i s a _ s a emen o his p oposi ion we conclude ha 8,,,(T) : N  N gi en P-LOCALIZATION OF SOME CLASSES OF GROUPS  25 by 0,, (7 - )  =  1N .2(T) . . . ¡(7-n- 1)  =  1NT . . . T n-1  is a bljec lon,  dT  E w(X), dn E P` . So, b'x E X V'n E P` cae ha e ha B,,(x) is a bijec ion, since B n (x) = 1N-w(x) . . . w(xn -1 ) = en(T), whe e T = w(x) E w(X) . Co olla y 1 .13 . Unde he condi ion o he p e ious p oposi ion (1 .12), G is P-local ~ N and X a e P-local . P oo .. See P op . 1 .9 . P oposi ion 1 .14 . Le N - G ~ X be an exac sequence o g oups . Then, G and X P-local ==> N P-local . P op . I ollows di ec ly om he de ini ions . a F om now on cae es ablish some esul s which play an impo an ole in Sec ion 3 . Le X -- w ~+ Au (A) and X - 0 -~ Au (B) be ac ions, whe e A and B a e abelian . Le also a E Homz[ X I (A, B) (ie a(w(x) .a) = 6(x) .ca(a)) . The eade in e es ed in mo e de ails abou he cons uc ions in ol ed in he p oposi ions below should collec ma e ial in [5, chap e II, P opo- si ion 4 .3 .], o ins an e . The p oo s o he nex h ee p oposi ions ollow easily om he de i- ni ions acco ding he usual echniques . P oposi ion 1 .15 . Conside he diag am whe e A and B a e abelian and he ocas a e exac . I he e exis s ,3 E Hom(G, Q) making he diag am commu a i e, hen a E Homz[XI (A, B) and a * l = 7*( . Con e sely, i a E Homz[X] (A, B) and a * j = , y*(, hen he e does exis ,l E Hom(G, Q) making he diag am commu a i e . P oposi ion 1 .16 . In he diag am G A Ñ G  X « 1  T ,[ .L N  1 -Y, X 1 ~ y 2 6  A . REYNOL FILHO he ows a e exac , A and B a e abelian and T and ~ yield commu a i i y . Then, he e exis s a c oss homomo phism : X --> B such ha dg E G, O(g) = ms(g) .T(g) . P oposi ion 1 .17 . A al B w G X 1T 1w i Q Y In he commu a i e diag am he ows a e exac and A and B abelian . Le X  "' i B be a c oss homomo phism . In hese condi ions he unc- ion G  0  Q gi en by l(g) = mE(g) . T (g), `dg E G is a g oup homomo - phism . Lemma 1 .18 . Le Q be a P- o sion abelian g oup . Then H < ,(Q) is a P- o sion abelian g oup, = dq > 0 . (He e Hq(Q) means he homology o he g oup Q wi h in ege coe icien s) . P oo .. The asse ion is eadily checked, since, acco ding he heo y in [2] we ha e Hn(Q)p, - H  ,(Qp,) = Hn((0)) = (0) ; n >_ 1 (once Q is P- o sion abelian) . Lemma 1 .19 . Le N  G  Q be a cen al exac sequence o g oups . I G ac s P-locally on an abelian g oup A, hen Q ac s P-locally on H* (N ; A) . P oo . We ecall ha i G ac s on A by means o w, hen he ac ion w is P-local i and only i (dn E P`) (Vx E G)B n(x) = lA +w(x) + + w(xn -1 ) E Au (A) . Mo eo e , aking z E G such ha E(z) = x, i is known ha he induced ac ion o Q on H'(N ; A) is gi en by : Q Au (H 9 (N ; A)), whe e Q(x) = w(z) * .( emembe ha he ex ension N , G -'» Q is cen al) . Thus, ixing x E Q and pu ing 6n (X) = 1Hs(N ;A) + 9 (x) +  + SZ(xn -1 ), we ge : '9n(X) = (1A)*+w(z)*+- . .+w(zn-1)* = [lA+w(z)+ . . . + w(zn -1 )] * = On(z)* . So On(x) is an isomo phism . Lemma 1 .20 . Suppose ha he ac ion X -- w --> Aú (A) is P-local and X is a P'- o sion g oup (A an abelian g oup) . Then, w is i ial . P oo .- Se x E X . By hypo hesis, 3 n E P'x such ha xn = 1 . SO, 0 = w(xn) - lA = 6 n (x)o[w(x) - lA] . P-LOCALIZATION OF SOME CLASSES OF GROUPS  2 7 Then, w(x) = lA ( o 0 n ,(x) E Au (A)) . The nex heo em is s a ed in he ca ego y 97 . Theo em 1 .21 . Conside he commu a i e diag am X ~ Au (A) whe e X is a nilpo en g oup, A is a P-local ini e abelian g oup and w, wp a e ac ions . 91 Then we ha e Hj p (Xp ; A)  HW (X ; A) . e o P oo .. (Induc ion on c = nil X .) I X is abelian, we ake he ollowing sho exac sequences : 0 -> Ke (eo) ____> X eó, eo(X) -> 0 . . . (1) 0 -> eo(X) - e, -> Xp ---> Coke (eo) -+ 0 . . . (2) (1) yields a spec al sequence (Lyndon-Hochschild-Se e) whe e E2 , s = H'(eo(X ) ; H'(Ke (eo) ; A)) . No icing ha Ke (eo) ac s i ially on A we a e allowed o say ha he ollowing sequence is exac 0 --> Ex (H S _1(Ke (eo)) ;A) > Hs(Ke (eo ;A)) ---> Hom(H,(Ke (eo) ;A) -~ 0 . Since Hom(P'- o sion, P-local) = (0) = Ex (P'- o sion, P-local) and (ds >0)H s (Ke (eo)) is P'- o sion (Lemma 1 .18) we conclude ha E2's = (0), ds > 0, whence he spec al sequence collapses . Thus, H , (eo (X) ; A) = E2'o = E , - H,,, (X ; A) . The e o e, we ha e go ha eó is an isomo phism . Likewise,  (2)  yields  ano he  spec al  sequence  whe e  E 2 > s  = H (Coke (eo) ; H'(eo(X) ; A)) .  He e  Coke (eo)  ac s  i ially  on H - '(eo(X ) ; A) . This may be seen om Lemmas 1 .19 and 1 .20 acco ding o he ol- lowing a gumen : due o P oposi ion 1 .5 wp(Xp) is a ( ini e) P- o sion g oup . So, Y = A 1 iwp(Xp) (whe e wp(Xp) y Au (A)) is a ini e P-g oup . So Y is P-local, and hen i ollows ha wp ac s P-locally on A (use he same a gumen ha he one in 1 .12) . Now, by Lemma 1 .19, we ha e ha Coke (e o ) ac s P-locally on Hs(eo(X) ;A) . As Coke (eo) is P'- o sion, ou s a emen now ollows om Lemma 1 .20 . 34  A . REYNOL FILHO Finally, we analyse he si ua ion in which A is (only) a ini e abelian g oup . Le X "-> Au (A) be an ac ion, whe e A is a ini e abelian g oup . I ¡ A l= pá' . . . p  , hen A i = pi-p ima y componen and Au (A) - l Au (A i ) . i=I Thus, he e is (uniquely de e mined) wi : X ---> Au (A i ) ; i = 1, . . . . Le G=A  W X ; G i =A i X ; G j> X ; G i ~> X as usual . I is well-known ha E is he pull-back o (Ei)1<i< - Le G  Gi be he p ojéc ion and G ~ Xp be he pull-back o he (EjP a ows (Gi) p --) Xp ; i = 1,  . . . Since (E_i)p o (Qj)p = 1X P , he e does exis (only one) Q E Hom(Xp, G) such ha ~ i o Q = (ui) p, Vi (he e Wi is he usual p o- jec ion) . Likewise, i is plain ha 3 ! E Hom(G, G) such ha ~ io = ei~i(ei Gi -) (Gj)p P-localizes Gi) . I ollows ha é = eOE and = Qeo . We ecall ha G is P-local by p op . 1 .1 . Mo eo e , 3  E Hom(Gp,G) such ha 7 ioo = (7 i)p, Since (Ei)po(7 i)p = Ep,,di (In pa icula 7 i is an isomo phism) . By unique ness we ha e go = Oe ; i~o = ep ; OQp = and E  = 1X P as well . (So G=CIXP) . Finally, le C = ke  - ® ke (Ei)p ; i=1 Ti :C , G,N _ =ke Ep ;p' :N-~Gp,e, ,0,de ine e :A->N, A -> C and i : N ---> C by es ic ion . Le B= ke 0 . Soon we a e going o show ha 3 ! e' E Hom(C, N) such ha e'7 = é . We a e able, a las , o cons uc he ollowing commu a i e diag am : whe e P-LOCALIZATION OF SOME CLASSES OF GROUPS  35 B K=Ke Diag am 2 .6 Lemma 2 .7 . is an epimo phism . P oo . This ollows om he ac ha E K = ® K¡ (K¡ = ke i-1 In o de o jus i y all he indica ions in he diag am, we s ill need wo lemmas . ke (Ejp N (Gilp ~-» Xp aken in he conjunc ion wi h he cases p e iously analysed . Lemma 2 .8 . é IK = 0 . P oo . Since 36  A . REYNOL FILHO we jus ha e o show ha é 1x ;= 0, Vi . This ollows om he diag am ( i is a spli ing a ached o S i) K i ~-a  Ai  Ñ  Gi  ~i'  G el  1 el  1 e i 1 , 1  ( )P ke (Ei)P ,--, (Gi)P  GP So we ha e e' E Hom(C, N) wi h e' = é . Lemma 2 .9 . (i)7(W(x) .a) =W(eo(x)) . (a)_  ;dx E X ;b'a E A (ii)é(w(x) .a) = wp(eo(x)) .e(a) } P oo . Bo h s a emen s a e eadily checked om he de ini ions . Theo em 2 .10 . In he condi ions abone, G -- -~> G P-localizes G . P oo : Le  Gp de ined by 0(p(c) .~j ;(z)) = p'e'(c) .ap(z) ; cE c l zEXp . 0 E Hom(G, Gp) by p op . 2 .2, so ha i is plain ha 3 . P-localiza ion on he ca ego y C Th oughou his sec ion we cons uc he heo y o P-localiza ion o a g oup in he ca ego y C o g oups which a e ex ensions o nilpo en g oups by ini e abelian g oups . (al hough we s ill use he same no a ion G ~ Gp o P-localiza ion in he ca ego y C) . P oposi ion 3 .1 . Le A ~ G ~ X be an exac sequence o g oups, whe e A is abelian ini e and X is nilpo en . Le X _ w -> Au (A) be he ac ion a ached o he ex ension and suppose I'W = A . Le also ,0  E  Hom(G,K) and B  Ñ  K  -'»  Y be an exac se quence, whe e B is ini e abelian and Y nilpo en .  Then, he e exis a E Hom(A, B) and ,y E Hom(X, Y) which yield commu a i i y in he diag am A al B X 1y Y P-LOCALIZATION OF SOME CLASSES OFGROUPS  37 P oo . Le H = op(A) < Y . I ollows om I` = A ha H C [Y, H] . So H C [Y, H] C I ,2 Y and hen, by induc ion, H C I' k y, bk >_ 2 ; whence H = {1} since Y is nilpo en . This comple es he p oo . P oposi ion 3 .2 . dG E l C 1, 3 ! U = U(G) < G, U ini e abelian wi h G/U nilpo en such ha F ue , = U, p o ided ha w is he ac ion a ached o he ex ension U >-~ G -» G/U . P oo - Le A >'-"-> G » X be an ex ension whe e A is ini e abelian and X is nilpo en . Le SZ : X -> Au (A) be he ac ion a ached o his ex ension, and se F = F', whe e is he smalles posi i e in ege such ha F = I' 1 . Le U= la(F) <G . So U is ini e abelian . Fu he mo e, A/F >-> G/U -» X is exac and X ac s nilpo en ly on A/F, so ha G/U is nilpo en . I 's also plain ha F u e , = U, i w(gU)u = gug -1 . Finally we poin ou ha he uniqueness ollows in a s aigh o wa d way om p oposi ion 3 .1 . Now le p be a p ime and C p be he ull sub-ca ego y o C o all g oups, which a e ex ensions o X by A, whe e A is a ini e abelian p-g oup . Co olla y 3 .3 . G E 1 C p 1 => U= U(G) is a ini e abelian p-g oup . P oo . In ac , U = u(F) and F is a sub-g oup o A . a Co olla y 3 .4 . G ESC I ; G is nilpo en ~ U= U(G) = {1} . P oo . (==) G E¡ l 1==> w : G/U ---> Au (A) is nilpo en ==> U = F2 _ . . . = FW 1 = {1} .(c = nil w) . (4--=) I is ob ious . Co olla y 3 .5 . G E l p  11 p, q p imes, p qL q, such ha G E 1 C p 1 n ¡C .1 . P oo - I ollows om co . 3 .3 and co . 3 .4 . We now de ine Gp El C 1 p o ided G El C 1 . Fix ~ : U >'-> G -'» G/U whe e U = U(G) is de ined by p op . 3 .2 ; w(gU)u = gug -1 and G/U (G/U)p P-localizes G/U in 77 . We conside 3 cases : Le p be a p ime and suppose i s ly G E l C p 1 . I) p E P' . Se e = eo o e, G » G/U Pa' (G/U)p . 38  A . REYNOL FILHO Then we ha e : HW(G/U ; U) - -'-* -H"-,(G/U ; U/I') So we mus conside (0) - (GIU)P = (G/U)P We should poin ou ha 7 * j = e*I P = 0 . II) pEP . Le P l = {q E P' : q 1 1 w(G/U)1 1},H =<x E G/U : o(w(x))Pj >,F = F(H) and w : G/U ----> Au (U/F) as de ined jus a e heo em 1 .21 . Co olla y 1 .25 allows us o claim ha 3 ! ac ion wP making commu a i e he diag am Gl u -i  Au (U/ ) / AI P (Glu)P Taking he na u al p ojec ion U --  U/F, we ha e ha 3 ! e*I P = 7 * l whe e Hwp((G1U)P ; U/F) Once mo e i is shown by p op . 1 .15 ha he e is a commu a i e diag am U > P + G  Glu le leo IP  U/ ,--, G P --  (Glu)P ¡el U lCPw o . p bP such ha A his poin i is impo an o poin ou ha we ha e de ined G E Up 1 C 1-+ G P E l C 1 and his de ini ion is "good" since G E l C p 1 n 1 C q ¡==¿- G E 17 (co . 3 .5) and hen U= {1} (co . 3 .4) In pa icula , his cons uc ion ex ends he one made in [2] . Example 3 .6 . Le w : 7L --> Au (Z/3 ® 7G/5) gi en by w(1) .a = 2a and w(1) .b = 2b . Le G= (Z/3 ® 7G/5) 'j ~,7Z . Then F2 = Z/3 ® 7G/5 = A, whence G 11 17 1 . Howe e G E l C 1 and since U= c(A), i ollows ha G 1 Up 1 C p 1 . This example shows ha P-LOCALIZATION OF SOME CLASSES OF GROUPS  39 III) G OCI Up ICpI- Now U is no longe a P-g oup . Ne e hless, Also, and es = (wl, .. . , wl) . We ha e 1 E H 2 (Gl u ; U) SP E H,,p(G/u)P ; U) whe e is de ined by (I) o (I1) . Also, whe e Ui is he pi-p ima y componen o U . G/U ---> Au (U) =  Au (Ui) i=1 (7 i . ,. . ,7 * ) i-1 whe e U -H Ui is he usual p ojec ion . No ice ha I'2 =U -4 I' 2 w  wi = U¡ ; Vi = 1, . . . , . Le i = S i * l and conside he commu a i e diag am (1 1* ,. . . ,7 - ) (7 1 * , . . . ,7 * ) Si Ui %+ Gi -» G/U Pi  1 Pi  Po ( i)P :  Ui  >li-x>  (Gi)p  ( P  (G/u)P (Si)i E ®í-1 H~2 i (Gl U ; Ui) ®x =1eo-1)(® z=1 Pi*) (Si)P E ®i=1H(w ;)n((G/U)P ; Ui and 3 ! Sp such ha  ,7F **)1p = ((ji)p)i p o ided ha  i is he usual p ojec ion and p = ®ipi . 40  A . REYNOL FILHO Diag am 3 .7 By de ini ion, we ha e ha IP is he pull-back o he a ows since As o G -j Gp de ined by (I) ;(II) ;(III) we ha e he nex wo p opo- si ions . P oposi ion 3 .8 . e is P-su jec i e . P ooi Ob ious . Suppose we ha e P oo . I ollows di ec ly om he de ini ions . Co olla y 3 .9 . G -- e > Gp --- 9 --> K, wi h K P-local . Then e = ge ==> = g . P oposi ion 3 .10 . G P-local == :> e is an isomo phism . P oo . We ha e 3 cases o analyse . The only one which is no ob ious is (II) . U  G -» G/ l i 1  1 .e  leo ,-- UI , GP -~ (Glu)P G P-local =>G/U P-local (co . 1 .9) . So w(G/U) is a P- o sion sub-g oup o Au (U), since Au (U) is ini e and G/U is P-local (co . 1 .3) . The e o e H = {1} and F = {l} . The e o e n = lu . So e is an isomo phism . P-LOCALIZATION OF SOME CLASSESOF GROUPS  41 Nex we conside a commu a i e diag am U(G) =  U  G  G/U la la 17 U(K) =  V  K  K/ Le us de ine á E Hom(U, V) induced by (apu =p a ; U P »U) . We ake U= ®U(p) and V = ®V(p) ; p-p ima y decomposi ions . P oposi ion 1 .26 assu es ha a(FU(p)) C FV(p), since a(U(p)) C V(p) . Ac ually, we conside ~(P)  U(p)  '--'  G(p)  - Gl U 1 alu(P) S(p)  V (P)  K (p)  -  K and hen use p op . 1 .26 o ~(p) _ 7 (p) *  and «p) = 7 (p) * ( . We de ine, by es ic ion, «p) : U(p) - V(p), whence we ha e PU(P) l U(p) =  U(p)/ u(P) and inally á = ®pd(p) . A his poin we s a e a undamen al p oposi ion . U(p)  a~P)  V (P) l PV(P) V (p)/FV(P) = V (P) P oposi ion 3 .11 . á is an homomo phism o modules . P ooL Le us conside he comu a i e diag ams . G U Au U) eo 1  / wa (G/U)P K -> Au (9) eo 1  /sap (Kl )P whe e wP and gP a e he ac ions gi en by he ex ensions U Ñ G P -» (G/U)P 1IX 17P V Ñ KP -» (K ) P 4 2  A . REYNOL FILHO We oug h o ge ha á(wp(z)Z) = Qp(yp(z)) .á(a), bz E (G/U)p and baEU . Fix z E (G/U)p and á E U . G/U nilpo en  n E P` such ha zn = eo(x) . Then, á(wp(zn) .á) = U(wp(eo(x)) .-j) = á(w(x) .á) = a(w(x) .a) (by de ini ion) = p a(w(x ) .a) = p (9(y(x)) .a(a)) (a is a homomo phism o modules) _ Q(y(x)) .a(a) = gp(eo(y(x))) .5(a) = QPyp(zn) .á(á) . . . (*) . On he o he hand o(wp(z)) = m E Px, (p op . 1 .2), since (GJU)p is P-local and Au (U) is ini e . So, gp(yp(zn)m) .á(j) = U(WP(z n ) m .Q) _ Z!(á),  E U . The e o e, gp(yP(z m ) n )I IX(U) - 1 IX(U) . S ill, aking in o accoun ha in he exac sequence (p : V >--> Kp ~> (K/V) p ; Kp and (K/V)p a e P-local we can s a e ha _ Bn(yp(zm)) _ 1V + gp(i'P(z m )) + . . . + gp(yp(zm)n-1) E Au (V),dn E P'x and , YP(z m ) E (K1V)p . Thus Va - E U we ha e : 0 = á(¿í) - gp(yp(zm)n) .Cj(Q,) _ [les - pp(i y p(z m )) n ja(a) = en(Í y p(z m » 0 [ 1 V- Qp(yp(zm))] .a(Z¡) . The e o e, 5 ( - j) - QP(yp(z m )) .a(a) = 0, whence Pp(yp(z m » Iá((1) = 1 -a(5) . Finally 3 , s  E  9G such ha m + sn  =  1  (gcd(m, n)  =  1) . The e o e á(wp(z) . á) = á(WP(z n )s oWP(z m ) . j) = á(WP(zn)s .á) = gp(_YP(z n ) 3 ) . á(á) = gp(i'P(z n )s) . qP(7P(z m ) ) .5 (C 1 ) =S2p(yp(z)) .a(-j) . Theo em 3 .12 . G, K E I C 1 ; 3 ! /3p E Hom(Gp, Kp) yelding commu- a i i y in he diag am I : U  »  G  ~>  G/U Sp  >i~> Gp > (G1U)P V  >, IK  ~>  K/ V Cp 'IYp Sp : V  >~>  Kp  -"* P  (K/V)p P oo ' The uniqueness ollows om he co olla y 3 .9 :, Fo he exis ence we obse e ha eóyP~p = .y*e*(p = y * p " ~ (de i- ni ion o (p) = p " y * ~ = p " a*(1) (p op . 1 .16) = á * pu* = ce * e*Ip (de . o ~p) = e*U*jp . I ollows ha y*~p = c** jp due o he ac ha H Z ((G/U)P ; V) --° > P-LOCALIZATION OF SOME CLASSES OF GROUPS  43 H 2 (G/U ;V) (Th . 1 .21) . So by p oposi ion 1 .16, 3 E Hom(Gp, Kp) yielding commu a i i y in he " on ace" o he diag am . Thus Te and eO make commu a i e he diag am ¿  e U GGl u p oa 1  e U e/3  1 eo V ,--, KP - (K1 )P Use o he p oposi ion 1 .17 shows ha 0 : G/U  > V a c oss ho- momo phism such ha e l(g) = 0e(g) .Te(g),dgE G . Howe e , HI((G/U)p ;V)  el . HI(G/U ; V) (Th . 1 .21) . So 9 = 0' eO + S , whe e w (x) = - x . , E V . Se ing S : (G/U)p  > V, 6 (z) = -z . , i ollows ha S o eO = S and he e o e 0 = Opoeo, whe e Op = BP + 6 . Now eO(g) = 9Peoe(g) .Te(g) = -POpEpe(g) .Te(g), dg E G . Thus de ining ,QP : Gp -> Kp by ~3p(z) = 0pep(z) .- (z),dz E Gp, i ollows om p op . 1 .18 ha ~3p E Hom(Gp,Kp) and ~3pe = eO . Besides, p,QP = - ypEP and Op7! = U . Rema k . The heo em abo e shows us ha G ---~ Gp is a unc o and e is a na u al ans o ma ion o unc o s . Theo em 3 .13 . G e > Gp P-localizes G in C . P oo . Le G, K E!¡ C 1, wi h K P-local, and 0 E Hom(G, K) . Owing o p oposi ion 3 .1, he e exis s a commu a i e diag am U(G) =  U  G  Gl la 10 17 U(K) =  V  >~>  K  ~>  Kl Now using h . 3 .12 we conclude ha 3 ! ~ 3p E Hom(Gp, Kp) such ha Ope = ei . So i is enough o ake  = e -1 o OP Gp - Kp Ap (p op . 3 .10) The uniqueness ollows om co . 3 .9 .