P-localization of some classes of groups

Publicacions Matemátiques, Vol 37 (1993), 19-44 . Abstract P-LOCALIZATION OF SOME CLASSES OF GROUPS AUGUSTO REYNOL FILHO The aim for the present paper is to study the theory of PLocalization of a group in a category C such that it contains the category of the nilpotent groups as a full sub-category . In the second section we present a number of results on P-localization of a group G, which is the semi-direct product of an abelian group A with a group X, in the category G of all groups . It tums out that the P-localized (Gp) is completely described by the P-localized Xp of X, A and the action w of X on A . In the third section, we present the construction of the theory of P-localization in the category of all groups which are extensions of nilpotent groups by finite abelian groups . Our proof follows rather closely the one presented in [2, chapter II, and is based on the classical interpretation of the second cohomology group of a group . Introduction Since Sullivan first pointed out the availability and applicability of localization methods in homotopy theory, there has been considerable work done on further developments and refinements of the method and on the study of new areas of application . In [2] P . Hilton, G . Mislin and J . Roitberg constructed the theory of P-localization of nilpotent groups, where P is a set of primes . Some time later, P . Ribenboin in [3] showed that it was possible to localize any group . (There is another approach concerning P-localization in group theory developed by Bousfield in Topology 14 (1975) 133-150, and Mem . Amer . Math . Soc . 1 0 (1977) no . 186, but, in this work, we just use the concepts presented in [2], [3] and [4]) . The construction presented in [3], however, seems to be quite abstract and this led us to try to obtain a more explicit construction of the P-localization of a group G in the category of all groups . We were successful when G is a semi-direct product of a finite abelian group A by 2 0  A . REYNOL FILHO some other group X . In addition, we managed to construct theory of Plocalization of a group in the category C of groups which are extensions of nilpotent groups by finite abelian groups . The question concerning semi-direct product is taken up in Section 2and the main results are 2 .1, 2 .5 and 2 .10 which could be stated as follows . Let P' be the complementary of P in the set of all primes . Theorem (2 .1) Let N -±-> G -'» X be an exact sequence of groups, where N is a p-group and p E P' . Then, e = eooEP-localizes G, provided that X  Xp P-localizes X . In this context, Theorem 2 .5 says the following : Let X  w ) Aut(A) be an action, where A is a finite abelian p-group and p E P . Let P l = {q E P' : q ~~w(X) ~} and consider Pi the multiplicative set generated by P, . Set H the sub-group of X generated by all x E X such that the order of w(x) belongs to P,' . Let wH be the restriction of w to H and F = l'F7, where r is the smallest positive integer such that F'H = I"Hl (Here Fi has the ordinary meaning and its definition may be found in [2]) . There is an action X  Aut(A/r) induced by w, which can be factored as X  w )  Aut(A/r) Let G = Al ~,X and G' = A/ r 1 W ,Xp Then, e : (a,, x) E G ---> (a+ F, eo(X)) E G' P-localizes G . Finally, Theorem 2 .10 analyzes the situation in which A is a finite abelian group . Let X ~ w _-> Aut(A) be an action, where A is a finite abelian group . Let A l . . . . , A t be the p-primary componente of A and wi : X ---> Aut(Ai) be the actions induced by w . - Let G = A ] c~X,Gi = A 2 ]  ,X and take G ~ E Xp to be the pull- (EJp back of the arrows (Gi) p  -*  Xp . (Notice that (Gi) p are given by either 2 .1 or 2 .5 . Then, we claim that the natural homomorphism G f G P-localizes G . We devote Section 3 to present our resulte concerning the construction of the theory of P-localization of a group in the category C of groups P-LOCALIZATION OF SOME CLASSES OFGROUPS  2 1 which are extensions of nilpotent groups by finite abelian groups . The matter could be described as follows : Given G El C 1, there exists a unique finite abelian sub-group U of G such that G/U is nilpotent and I'2 =U where W is the action attached to the extension : U N G -» G/U . Furthermore, there exists a unique group U and an epimorphism pv : U -» U and a unique bP E H Z ((G1U)P ; i7) . (bp : U - GP - (Gl u)P) attached to 1 yielding commutativity in the diagram U G Gw I PU  f e  l eo jP :  _  ,--,  G P  -~  (Gw)P Under such conditions (3 .12) states that G  > G P is a functor and e is a natural transformation of functors . In addition, (3 .13) also states that eP-localizes G . In Section 1 we introduce some basic results needed in the following section . We believe, nevertheless, that Theorem (1 .21) is important on its own accord ; it states that if X -~ Aut(A) is a comutative diagram where X ~- e-'> X P P-localizes X and A is a Plocal finite abelian group, then eo* : H,1 P (X P ; A)  H, 1 (X ; A) is an isomorphism . This work is the main part of the author's Ph .D . thesis done under the guidance of Professor Peter John Hilton . The author is very much indebted to Professor Daciberg Lima Gonjalves, at whose suggestion this work was developed . 1 . Preliminaries In this section we introduce some general results on P-local groups, factoring of actions and some propositions concerning group cohomology . We start by fixing the notations P' = {n E N* : p 1 n => p E P} = multiplicative set generated by P ; P' is the complementary of P in the set of all primes . We also recall that G is said to be a P-local group  ((dn E P'm)x E G ~--> x' E G is bijective) . 22  A . REYNOL FILHO Moreover, G  e ) GP P-localizes G E l C 1 in the category C 4=>(GP E IC 1, GP is P-local and b'H E I C 1, H P-local, (V f E HOm(G, H»(3! f p E Hom(GP, H)) which yields commutativity in the diagram G --~, H el  lfp  ) . G P Proposition 1 .1 . Let E,, ... , E t , K be P-local groups, where P is a set of primes . Let el E Hom(Ej, K), i = 1, . . . , t . In these conditions, if E E Hom(E, K) is the pull-back of the family (Ej)I<j<t, then E is P-local . Proof . Straightforward . Proposition 1 .2 . Let 0 E Hom(Y F), where Y is a P-local group and F is a finite group . Then, (dy E Y) we have o(O(y)) =n E Px ; (o(O(y)) = order of O(y)) . Proof .. Let yE Y and suppose that 3q E P' with q 1 o(O(Y)) . Then we may consider z = y k , where o(o(y)) = q .k . It follows that o(o(z)) = q . The fact that Y is P-local and q E P' enables us to state that dr > 0, 3zr E Y such that z4 r = z . Thus, O(zr)gr = O(z) ~ 1 and 4'(zr)gr+' - O(z) 4 = 1, SO o(O(zr)) = qr+l, yr > 0 . In particular {O(zr) E F : r > 0} is infinite . However this is impossible, since F is finite . Corollary 1 .3 .  Under the conditions of the previous proposition, we have 1 O(Y) ¡ E Px (ie, O(Y) is a P-torsion (finite) sub-group of F) . Remark 1 .4 . According to proposition (7 .1) in [4] we have that a finite group F is P-local  F is a P-torsion group . The full subcategory of the category 9 of all groups consisting of all nilpotent groups is denoted by rl . Proposition 1 .5 . Let X -- w -> Aut(N) be an action, where X and N are groups with Aut(N) finite . Then 3 !wp such that the diagram X Aut(N) eo i XP is commutative  w(X) is a P-torsion sub-group of Aut(N) . Proof . It follows directly from Corollary 1 .3 and Remark 1 .4 . P-LOCALIZATION OF SOME CLASSES OF GROUPS  2 3 Now we take A Ñ G -'!» X an exact sequence of groups, where A is abelian . Let X -- w -> Aut(A) be the action defined by p(w((x) .a) = g.p(a) .g-1 (where E (g) = x) . Fix a collection of primes P, n E N and x E X, and define : en(x) = lA + w(x) + . . . +w(xn -1 ) E End(A) . For this endomorphism we have : Lemma 1.6 . (,u(a) . g) n = p(0(E(g)) .a)gn ; b'g E G, da E A, dn E N . ProoE It is easy by induction on n . Proposition 1 .7 . Let P be a set of primes and let A ~ G ~ X be an exact sequence of groups, where A is abelian and w is the action attached to the extension . Fix the conditions : (i) G is P-local ; (ii) X is P-local ; (iii) 9 n (x) E Aut(A),Vx E X,`dn E P" . Then, if two of (i) ; (ii) ; (iii) ; hold so does the third . Proof .. (ii) + (iii) ==> (i) . Fix n E P" . Let g, h E G and suppose that 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, let g E GAx E X such that E(g) = xn (XP-local) . Therefore, e(g) = xn = E(hn) . Therefore g = u(a) .hn . Take b E A such that a = Bn(e(h)) .b . Thus g = F¿(B,(E(h)) .b) .hn = (p(b) .h)n due to 1 .6 . So g E G  > gn E G is bijective . (The other implications are similar) . Proposition 1 .8 . Let P be a set of primes and let A Ñ G  X be an exact sequence of groups, where A is finite abelian .  Then, Bn(x) E Aut(A), dx E X, dn E P", provided that either G is P-local or A and X are P-local . Proof . (I) G is P-local . Fix n E P` and x E X . Suppose that Bn (x) .a = 0 . Let g E G such that 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 Aut(A) since A is finite . (II) A and X are 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 therefore w(x') .a = a . On the other 2 4  A . REYNOL FILHO hand, o(w(x)) = m E P' (Prop . 1 .2) . Therefore w(x)m .a = a . As gcd(m, n) = 1, it follows that w(x) .a = a, whence 0 = B n (x) .a = n .a . Thus a = 0 since A is P-local . So 8,,,(x) E Aut(A) . a Corollary 1 .9 . In the conditions of the proposition aboye (1 .8), G is P-local <-==~ A and X are P-local . E - "I Next we consider a split extension N >4 G  X, where N is a fi0 nite group . Let X --- w -> Aut(N) be the action given by p(w(x) .a) = u(x) .p(a) .o,(x)-1 and take Bn(x) : N --- N defined by B n (x) .a = 1A(a) .(w(x) .a) . . . (w(x)n -1 . a), x EX ; n EN* : In this slightly different context we now describe properties which are quite similar to Prop . 1 .6, 1 .7, 1 .8 and 1 .9 . Lemma 1 .10 . (p(a) .(x))n = P(v n (x) .a) o . (X)n ; dx E X ; dn E N* ; `da E A . Proof . See Prop . 1 .6 . Proposition 1 .11 . Let N Ñ G <--< X be a split short exact sequence of groups and let w be the action defined by the splitting o .  Fix the statements : (i) G is P-local ; (ii) X is P-local ; (iii) en(x) is a bijection, bx E X ; `dn E P'x . Then, if two of (i) ; (ü) ; (iii) ; hold, so does the third . Proof . See Prop . 1 .7 . E Proposition 1 .12 . Let N >-~ G E--< X be a split short exact sequence 0 of groups, where N is finite . Then Bn(x) is a bijection, dn E P", dx E X provided that either G is P-local or N and X are P-local . Proof . If G is P-local, then the proof follows as (I) Prop . 1 .8 . So let's suppose that N and X are P-local . Let w(X) ~- i > Aut(N) and G= N 1 Zw(X) . w(X) is a P-torsion group (Cor . 1 .3) and N is a P-torsion group (Remark 1 .4) . Thus G is a P-torsion group . So G is P-local (Remark 1 .4) . E Then taking the sequence N ~ G <Z w(X) and invoking the first a _ statement of this proposition we conclude that 8,,,(T) : N  N given P-LOCALIZATION OF SOME CLASSES OF GROUPS  25 by 0,, (7 - )  =  1N .2(T) . . . ¡(7-n1)  =  1NT . . . T n-1  is a bljectlon,  dT  E w(X), dn E P` . So, b'x E X V'n E P` cae have that B,,(x) is a bijection, since B n (x) = 1N-w(x) . . . w(xn -1 ) = en(T), where T = w(x) E w(X) . Corollary 1 .13 . Under the condition of the previous proposition (1 .12), G is P-local ~ N and X are P-local . Proof .. See Prop . 1 .9 . Proposition 1 .14 . Let N - G ~ X be an exact sequence of groups . Then, G and X P-local ==> N P-local . Propf . It follows directly from the definitions . a From now on cae establish some results which play an important role in Section 3 . Let X -- w ~+ Aut(A) and X - 0 -~ Aut(B) be actions, where A and B are abelian . Let also a E Homz[ X I (A, B) (ie a(w(x) .a) = 6(x) .ca(a)) . The reader interested in more details about the constructions involved in the propositions below should collect material in [5, chapter II, Proposition 4 .3 .], for instante . The proofs of the next three propositions follow easily from the definitions according the usual techniques . Proposition 1 .15 . Consider the diagram where A and B are abelian and the rocas are exact . If there exists ,3 E Hom(G, Q) making the diagram commutative, then a E Homz[XI (A, B) and a * l = 7*( . Conversely, if a E Homz[X] (A, B) and a * j = , y*(, then there does exist ,l E Hom(G, Q) making the diagram commutative . Proposition 1 .16 . In the diagram G A Ñ G  X « 1  T ,[ .L N  1 -Y, X 1 ~r y 2 6  A . REYNOL FILHO the rows are exact, A and B are abelian and T and ~ yield commutativity . Then, there exists a cross homomorphism r : X --> B such that dg E G, O(g) = ms(g) .T(g) . Proposition 1 .17 . A al B w G X 1T 1w ir Q Y In the commutative diagram the rows are exact and A and B abelian . Let X  "' i B be a cross homomorphism . In these conditions the function G  0  Q given by fl(g) = mE(g) . T (g), `dg E G is a group homomorphism . Lemma 1 .18 . Let Q be a P-torsion abelian group . Then H < ,(Q) is a P-torsion abelian group, = dq > 0 . (Here Hq(Q) means the homology of the group Q with integer coeficients) . Proof .. The assertion is readily checked, since, according the theory in [2] we have Hn(Q)p, - H  ,(Qp,) = Hn((0)) = (0) ; n >_ 1 (once Q is P-torsion abelian) . Lemma 1 .19 . Let N  G  Q be a central exact sequence of groups . If G acts P-locally on an abelian group A, then Q acts P-locally on H* (N ; A) . Proof . We recall that if G acts on A by means of w, then the action w is P-local if and only if (dn E P`) (Vx E G)B n(x) = lA +w(x) + + w(xn -1 ) E Aut(A) . Moreover, taking z E G such that E(z) = x, it is known that the induced action of Q on H'(N ; A) is given by : Q Aut(H 9 (N ; A)), where Q(x) = w(z) * .(remember that the extension N , G -'» Q is central) . Thus, fixing x E Q and putting 6n (X) = 1Hs(N ;A) + 9 (x) +  + SZ(xn -1 ), we get : '9n(X) = (1A)*+w(z)*+- . .+w(zn-1)* = [lA+w(z)+ . . . + w(zn -1 )] * = On(z)* . So On(x) is an isomorphism . Lemma 1 .20 . Suppose that the action X -- w --> Aút(A) is P-local and X is a P'-torsion group (A an abelian group) . Then, w is trivial . Proof .- Set x E X . By hypothesis, 3 n E P'x such that 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 (for 0 n ,(x) E Aut(A)) . The next theorem is stated in the category 97 . Theorem 1 .21 . Consider the commutative diagram X ~ Aut(A) where X is a nilpotent group, A is a P-local finite abelian group and w, wp are actions . 91 Then we have Hj p (Xp ; A)  HW (X ; A) . e o Proof .. (Induction on c = nil X .) If X is abelian, we take the following short exact sequences : 0 -> Ker(eo) ____> X eó, eo(X) -> 0 . . . (1) 0 -> eo(X) - e, -> Xp ---> Coker(eo) -+ 0 . . . (2) (1) yields a spectral sequence (Lyndon-Hochschild-Serre) where E2 , s = H'(eo(X ) ; H'(Ker(eo) ; A)) . Noticing that Ker(eo) acts trivially on A we are allowed to say that the following sequence is exact 0 --> Ext(H S _1(Ker(eo)) ;A) > Hs(Ker(eo ;A)) ---> Hom(H,(Ker(eo) ;A) -~ 0 . Since Hom(P'-torsion, P-local) = (0) = Ext(P'-torsion, P-local) and (ds >0)H s (Ker(eo)) is P'-torsion (Lemma 1 .18) we conclude that E2's = (0), ds > 0, whence the spectral sequence collapses . Thus, H r, (eo (X) ; A) = E2'o = Er, - H,,, (X ; A) . Therefore, we have got that eó is an isomorphism . Likewise,  (2)  yields  another  spectral  sequence  where  E 2 > s  = Hr(Coker(eo) ; H'(eo(X) ; A)) .  Here  Coker(eo)  acts  trivially  on H - '(eo(X ) ; A) . This may be seen from Lemmas 1 .19 and 1 .20 according to the following argument : due to Proposition 1 .5 wp(Xp) is a (finite) P-torsion group . So, Y = A 1 iwp(Xp) (where wp(Xp) y Aut(A)) is a finite P-group . So Y is P-local, and then it follows that wp acts P-locally on A (use the same argument that the one in 1 .12) . Now, by Lemma 1 .19, we have that Coker(e o ) acts P-locally on Hs(eo(X) ;A) . As Coker(eo) is P'-torsion, our statement now follows from Lemma 1 .20 . 34  A . REYNOL FILHO Finally, we analyse the situation in which A is (only) a finite abelian group . Let X "-> Aut(A) be an action, where A is a finite abelian group . If ¡ A l= pá' . . . pt  , then A i = pi-primary component and t Aut(A) - rl Aut(A i ) . i=I Thus, there is (uniquely determined) wi : X ---> Aut(A i ) ; i = 1, . . . t . Let G=A  W X ; G i =A i X ; G j> X ; G i ~> X as usual . It is well-known that E is the pull-back of (Ei)1<i<tLet G  Gi be the projéction and G ~ Xp be the pull-back of the (EjP arrows (Gi) p --) Xp ; i = 1,  . . t . Since (E_i)p o (Qj)p = 1X P , there does exist (only one) Q E Hom(Xp, G) such that ~fi o Q = (ui) p, Vi (here Wi is the usual projection) . Likewise, it is plain that 3 ! f E Hom(G, G) such that ~riof = ei~i(ei Gi -) (Gj)p P-localizes Gi) . It follows that é f = eOE and f tr = Qeo . We recall that G is P-local by prop . 1 .1 . Moreover, 3  E Hom(Gp,G) such that 7rioo = (7ri)p, Since (Ei)po(7ri)p = Ep,,di (In particular 7ri is an isomorphism) . By unique ness we have got f = Oe ; i~o = ep ; OQp =v and E  = 1X P as well . (So G=CIXP) . Finally, let t C = ker  - ® ker(Ei)p ; i=1 Ti :C , G,N _ =kerEp ;p' :N-~Gp,e,f,0,define e :A->N,f A -> C and i : N ---> C by restriction . Let B= ker 0 . Soon we are going to show that 3 ! e' E Hom(C, N) such that e'7 = é . We are able, at last, to construct the following commutative diagram : where P-LOCALIZATION OF SOME CLASSES OF GROUPS  35 B K=Ker Diagram 2 .6 Lemma 2 .7 . f is an epimorphism . Proof . This follows from the fact that t E t K = ® K¡ (K¡ = ker i-1 In order to justify all the indications in the diagram, we still need two lemmas . ker(Ejp N (Gilp ~-» Xp taken in the conjunction with the cases previously analysed . Lemma 2 .8 . é IK = 0 . Proof . Since 36  A . REYNOL FILHO we just have to show that é 1x ;= 0, Vi . This follows from the diagram (vi is a splitting attached to Sri) K i ~-a  Ai  Ñ  Gi  ~i'  G el  1 el  1 e vti 1 , 1  ( )P ker(Ei)P ,--, (Gi)P  GP So we have e' E Hom(C, N) with e' f = é . Lemma 2 .9 . (i)7(W(x) .a) =W(eo(x)) .f(a)_  ;dx E X ;b'a E A (ii)é(w(x) .a) = wp(eo(x)) .e(a) } Proof . Both statements are readily checked from the definitions . Theorem 2 .10 . In the conditions abone, G -- f -~> G P-localizes G . Proof : Let  Gp defined by 0(p(c) .~j ;(z)) = p'e'(c) .ap(z) ; cE c l zEXp . 0 E Hom(G, Gp) by prop . 2 .2, so that it is plain that 3 . P-localization on the category C Throughout this section we construct the theory of P-localization of a group in the category C of groups which are extensions of nilpotent groups by finite abelian groups . (althought we still use the same notation G ~ Gp for P-localization in the category C) . Proposition 3 .1 . Let A ~ G ~ X be an exact sequence of groups, where A is abelian finite and X is nilpotent . Let X _ w -> Aut(A) be the action attached to the extension and suppose I'W = A . Let also ,0  E  Hom(G,K) and B  Ñ  K  -'»  Y be an exact se quence, where B is finite abelian and Y nilpotent .  Then, there exist a E Hom(A, B) and ,y E Hom(X, Y) which yield commutativity in the diagram A al B X 1y Y P-LOCALIZATION OF SOME CLASSES OFGROUPS  37 Proof . Let H = rop(A) < Y . It follows from I` = A that H C [Y, H] . So H C [Y, H] C I ,2 Y and then, by induction, H C I' k y, bk >_ 2 ; whence H = {1} since Y is nilpotent . This completes the proof . Proposition 3 .2 . dG E l C 1, 3 ! U = U(G) < G, U finite abelian with G/U nilpotent such that F ue , = U, provided that w is the action attached to the extension U >-~ G -» G/U . ProofLet A >'-"-> G » X be an extension where A is finite abelian and X is nilpotent . Let SZ : X -> Aut(A) be the action attached to this extension, and set F = F', where r is the smallest positive integer such that Fr= I'r 1 . Let U= la(F) <G . So U is finite abelian . Furthermore, A/F >-> G/U -» X is exact and X acts nilpotently on A/F, so that G/U is nilpotent . It's also plain that F u e , = U, if w(gU)u = gug -1 . Finally we point out that the uniqueness follows in a straightforward way from proposition 3 .1 . Now let p be a prime and C p be the full sub-category of C of all groups, which are extensions of X by A, where A is a finite abelian p-group . Corollary 3 .3 . G E 1 C p 1 => U= U(G) is a finite abelian p-group . Proof . In fact, U = u(F) and F is a sub-group of A . a Corollary 3 .4 . G ESC I ; G is nilpotent ~ U= U(G) = {1} . Proof . (==) G E¡ rl 1==> w : G/U ---> Aut(A) is nilpotent ==> U = F2 _ . . . = FW 1 = {1} .(c = nil w) . (4--=) It is obvious . Corollary 3 .5 . G E l p  11 p, q primes, p qL q, such that G E 1 C p 1 n ¡C .1 . ProofIt follows from cor . 3 .3 and cor . 3 .4 . We now define Gp El C 1 provided G El C 1 . Fix ~ : U >'-> G -'» G/U where U = U(G) is defined by prop . 3 .2 ; w(gU)u = gug -1 and G/U (G/U)p P-localizes G/U in 77 . We consider 3 cases : Let p be a prime and suppose firstly G E l C p 1 . I) p E P' . Set e = eo o e, G » G/U Pa' (G/U)p . 38  A . REYNOL FILHO Then we have : HW(G/U ; U) - -'-* -H"-,(G/U ; U/I') So we must consider (0) - (GIU)P = (G/U)P We should point out that 7r * j = e*I P = 0 . II) pEP . Let 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 ----> Aut(U/F) as defined just after theorem 1 .21 . Corollary 1 .25 allows us to claim that 3 ! action wP making commutative the diagram Gl u -i  Aut(U/r) / AI P (Glu)P Taking the natural projection U --  U/F, we have that 3 ! e*I P = 7r * l where Hwp((G1U)P ; U/F) Once more it is shown by prop . 1 .15 that there is a commutative diagram U > P + G  Glu le leo IP  U/r ,--, G P --  (Glu)P ¡el \U lCPw o . p bP such that At this point it is important to point out that we have defined G E Up 1 C 1-+ G P E l C 1 and this definition is "good" since G E l C p 1 n 1 C q ¡==¿- G E 17 (cor . 3 .5) and then U= {1} (cor . 3 .4) In particular, this construction extends the one made in [2] . Example 3 .6 . Let w : 7L --> Aut(Z/3 ® 7G/5) given by w(1) .a = 2a and w(1) .b = 2b . Let G= (Z/3 ® 7G/5) 'j ~,7Z . Then F2 = Z/3 ® 7G/5 = A, whence G 11 17 1 . However G E l C 1 and since U= tc(A), it follows that G 1 Up 1 C p 1 . This example shows that P-LOCALIZATION OF SOME CLASSES OF GROUPS  39 III) G OCI \Up ICpINow U is no longer a P-group . Neverthless, Also, and es = (wl, .. . , wl) . We have 1 E H 2 (Gl u ; U) SP E H,,p(G/u)P ; U) where is defined by (I) or (I1) . Also, where Ui is the pi-primary component of U . t G/U ---> Aut(U) =  Aut(Ui) i=1 t (7ri . ,. . ,7rt* ) i-1 where U -H Ui is the usual projection . Notice that I'2 =U -4 I' 2 w  wi = U¡ ; Vi = 1, . . . , t . Let fi = Sri * l and consider the commutative diagram (1r1* ,. . . ,7rt- ) (7r 1 * , . . . ,7r t * ) 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) ®xt=1eo-1)(®tz=1 Pi*) (Si)P E ®i=1H(w ;)n((G/U)P ; Ui and 3 ! Sp such that  ,7F t **)1p = ((ji)p)i provided that  i is the usual projection and p = ®ipi . 40  A . REYNOL FILHO Diagram 3 .7 By definition, we have that IP is the pull-back of the arrows since As for G -j Gp defined by (I) ;(II) ;(III) we have the next two propositions . Proposition 3 .8 . e is P-surjective . Prooi Obvious . Suppose we have Proof . It follows directly from the definitions . Corollary 3 .9 . G -- e > Gp --- 9 --> K, with K P-local . Then f e = ge ==> f = g . Proposition 3 .10 . G P-local == :> e is an isomorphism . Proof . We have 3 cases to analyse . The only one which is not obvious is (II) . U  G -» G/rl ir 1  1 .e  leo ,-- UIr , GP -~ (Glu)P G P-local =>G/U P-local (cor . 1 .9) . So w(G/U) is a P-torsion sub-group of Aut(U), since Aut(U) is finite and G/U is P-local (cor . 1 .3) . Therefore H = {1} and F = {l} . Therefore n = lu . So e is an isomorphism . P-LOCALIZATION OF SOME CLASSESOF GROUPS  41 Next we consider a commutative diagram U(G) =  U  G  G/U la la 17 U(K) =  V  K  K/v Let us define á E Hom(U, V) induced by (apu =pva ; U P »U) . We take U= ®U(p) and V = ®V(p) ; p-primary decompositions . Proposition 1 .26 assures that a(FU(p)) C FV(p), since a(U(p)) C V(p) . Actually, we consider ~(P)  U(p)  '--'  G(p)  - Gl U 1 alu(P) S(p)  V (P)  K (p)  -  Kv and then use prop . 1 .26 to ~(p) _ 7r(p) *  and «p) = 7r(p) * ( . We define, by restriction, «p) : U(p) - V(p), whence we have PU(P) l U(p) =  U(p)/ru(P) and finally á = ®pd(p) . At this point we state a fundamental proposition . U(p)  a~P)  V (P) l PV(P) V (p)/FV(P) = V (P) Proposition 3 .11 . á is an homomorphism of modules . ProoL Let us consider the comutative diagrams . G U AutU) eo 1  / wa (G/U)P K v -> Aut(9) eo 1  /sap (Klv)P where wP and gP are the actions given by the extensions U Ñ G P -» (G/U)P 1IX 17P V Ñ KP -» (Kv) P 4 2  A . REYNOL FILHO We ougth to get that 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 nilpotent  n E P` such that zn = eo(x) . Then, á(wp(zn) .á) = U(wp(eo(x)) .-j) = á(w(x) .á) = a(w(x) .a) (by definition) = pva(w(x ) .a) = pv(9(y(x)) .a(a)) (a is a homomorphism of modules) _ Q(y(x)) .a(a) = gp(eo(y(x))) .5(a) = QPyp(zn) .á(á) . . . (*) . On the other hand o(wp(z)) = m E Px, (prop . 1 .2), since (GJU)p is P-local and Aut(U) is finite . So, gp(yp(zn)m) .á(j) = U(WP(z n ) m .Q) _ Z!(á),  E U . Therefore, gp(yP(z m ) n )I IX(U) - 1 IX(U) . Still, taking into account that in the exact sequence (p : V >--> Kp ~> (K/V) p ; Kp and (K/V)p are P-local we can state that_ Bn(yp(zm)) _ 1V + gp(i'P(z m )) + . . . + gp(yp(zm)n-1) E Aut(V),dn E P'x and , YP(z m ) E (K1V)p . Thus Va - E U we have : 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 VQp(yp(zm))] .a(Z¡) . Therefore, 5 ( - j) - QP(yp(z m )) .a(a) = 0, whence Pp(yp(z m » Iá((1) = 1 -a(5) . Finally 3 r, s  E  9G such that rm + sn  =  1  (gcd(m, n)  =  1) . Therefore á(wp(z) . á) = á(WP(z n )s oWP(z m ) r . j) = á(WP(zn)s .á) = gp(_YP(z n ) 3 ) . á(á) = gp(i'P(z n )s) . qP(7P(z m ) r ) .5 (C 1 ) =S2p(yp(z)) .a(-j) . Theorem 3 .12 . G, K E I C 1 ; 3 ! /3p E Hom(Gp, Kp) yelding commutativity in the diagram I : U  »  G  ~>  G/U Sp  >i~> Gp > (G1U)P V  >, IK  ~>  K/ V Cp 'IYp Sp : V  >~>  Kp  -"* P  (K/V)p Proof' The uniqueness follows from the corollary 3 .9 :, For the existence we observe that eóyP~p = .y*e*(p = y * pv " ~ (definition of (p) = pv " y * ~ = pv " a*(1) (prop . 1 .16) = á * pu* = ce * e*Ip (def . of ~p) = e*U*jp . It follows that y*~p = c** jp due to the fact that H Z ((G/U)P ; V) --° > P-LOCALIZATION OF SOME CLASSES OF GROUPS  43 H 2 (G/U ;V) (Th . 1 .21) . So by proposition 1 .16, 3 r E Hom(Gp, Kp) yielding commutativity in the "front face" of the diagram . Thus Te and eO make commutative the diagram t¿  e U GGl u pvoa 1  re U e/3  1 eo V ,--, KP - (K1v)P Use of the proposition 1 .17 shows that 0 : G/U  > V a cross homomorphism such that efl(g) = v0e(g) .Te(g),dgE G . However, HI((G/U)p ;V)  el . HI(G/U ; V) (Th . 1 .21) . So 9 = 0' eO + Sv, where w (x) = v - x .v, v E V . Setting S : (G/U)p  > V, 6v(z) = v-z .v, it follows that S v o eO = Sv and therefore 0 = Opoeo, where Op = BP + 6v . Now eO(g) = v9Peoe(g) .Te(g) = -POpEpe(g) .Te(g), dg E G . Thus defining ,QP : Gp -> Kp by ~3p(z) = v0pep(z) .-r(z),dz E Gp, it follows from prop . 1 .18 that ~3p E Hom(Gp,Kp) and ~3pe = eO . Besides, rp,QP = - ypEP and Op7! = vU . Remark . The theorem above shows us that G ---~ Gp is a functor and e is a natural transformation of functors . Theorem 3 .13 . G e > Gp P-localizes G in C . Proof . Let G, K E!¡ C 1, with K P-local, and 0 E Hom(G, K) . Owing to proposition 3 .1, there exists a commutative diagram U(G) =  U  G  Glv la 10 17 U(K) =  V  >~>  K  ~>  Klv Now using th . 3 .12 we conclude that 3 ! ~ 3p E Hom(Gp, Kp) such that Ope = ei . So it is enough to take  = e -1 o OP Gp - Kp Ap (prop . 3 .10) The uniqueness follows from cor . 3 .9 .