On induced morphisms of mislin genera

Publicacions Matemàtiques, Vol 38 (1994), 299314 . ON INDUCED MoRPHISM S OF MISLIN GENER A PETER . HILTO N Dedicated to my good friend Karl Gruenberg, in admiration and affection , on the occasion of his 65 th birthda y Abstract  Let N be a nilpotent group with torsion subgroup TN, and le t a : TN —> T be a surjective homomorphism such that ker a i s normal in N . Then a determines a nilpotent group N such tha t TN = T and a function a * from the Mislin genus of N to that of N if N (and hence N) is finitely generated . The association a l—> oc * satisfies the usual functorial conditions . Moreover [N, N] is finit e if and only if [N, N] is finite and is then a homomorphis m of abelian groups . If Ñ belongs to the special class studied b y Casacuberta and Hilton ( Comm . irt . AZg . 19(7) (1991), 2051 = 2069), then cx . is surjective . The construction cx * thus enables u s to prove that the genus of N is non-trivial in many cases in whic h N itself is not in the special class ; and to establish noncancellatio n phenomena relating to such groups N . U . Introductio n Guido Mislin introduced and discussed in [M] the genus U ~N} of a finitely generated (f g ) nilpotent group N . This consists of isomorphis m classes of f g nilpotent groups M such tha t (0 .1)  M p N p , for allprimes p , where Mp is the p -localization of M . By abuse we say that M belong s to O(N) . It was early known that C~ ~N} is not trivial, but systemati c methods of calculating the set C~ ~N} and representing its elements wer e lacking . Mislin himself in [M], and together with the present author in [HM] , described an abelian group structure which could be introduced int o C3 ~N} if N satisfied the condition that its commutator subgroup [N, N] 300  P . HILTO N is finite ; we call the class of such f g nilpotent groups No ; moreover , ~ ~N} is then finite . However, this still did not permit any kind of systematic calculation of e3(N) . Calculations were done for specific group s in [H2] . Later, Casacuberta and Hilton [CH] introduced a class of nilpotent groups C No, and calculated ~~N} for N E N1 ; they furthe r showed how to modify N to realize any given element in O(N) . Th e nature of the groupsin N 1 was further analysed in [S], [HS1J indeed , the class is very strongly restricted— and, in [S], [HS2], the calculatio n of the genus was extended from N to N k , the direct product of k copie s of N, provided N E N 1 . A key result in this work is that, for N E N1 , U ~N} can only be non-trivial if FN = N/TN is cyclic, where TN is th e torsion subgroup of N ; recall that FN is commutative far N E No . A significant difficulty in attempting to calculate e5(N) is that lack s any kind of functoriality . We endeavor in this paper to go some wa y towards remedying this defect . Thus we suppose given a f g nilpoten t group N and a surjective homomorphism a : TN —} T, for some finit e group T which is, of course, necessarily nilpotent . Given the supplementary condition that ker a is normal in N, we construct a f g nilpoten t group N such that TN = T and a function cx . : ~ ~N} -> O(N) . Moreover, N E No if and only if N E No ; and a * is then a homomorphism . It is easy to see thata 1—> a * satisfies the usual functoriality conditions . Further we show in Section 2 that if N E N1 then a * is surjective ; thus , in this case, considerable information is made available about (N) , since we may calculate O(N) . A particular, and important, example of the construction is afforded by taking T to be the abelianization of TN with a the abelianizin g homomorphism . To avoid triviality we take FN cyclic . Then N satisfie s two of the three conditions for membership of J~ 1 (see below) . Moreover , the third condition will be automatically satisfied if T happens to b e cyclic . We also show in Section 2 that a non -cancellation result proved i n [CHI far groups in N I extends to groups, which, in our sense aboye, li e over groups in N 1. That is, we obtain pairwise non-isomorphic group s (L,M, . . .)in5(N)suchthatLxCMxCNxC,where C is cyclic infinite . In Section 3 we give a typical example of the application of the method , with explicit calculations . For the convenience of the reader, we collect here the crucial fact s about the class N1 . We assume N E No and refer to the extensio n (0 .2)  TN >--> N --H FN . MORPHISMS OF MISLIN GENERA  30 1 ThenNE]~ 1 i f (i) TN is commutative ; (u) (0 .2) is a split extension for an action w : FN —> Aut TN ; (iii) w(FN) lies in the center of Aut TN . We then note that, in the presence of (i), condition (iii) is equivalent t o (iii)' for each 1 E FN, there exists a positive integer u such that 1 . a = u a , for all a E TN . To avoid a trivial genus, we assume FN cyclic, say, FN = ( 0 . Let t be the order of w(e) in Aut TN . Then [CHI, if N E 911 , (0 .3)  (Z/t)*/{±1} . Moreover, if [m] E (7Z/t)*/{±1}, where m is prime to t, we may choos e the isomorphism (0 .3) so that the group corresponding to m is obtained from N by introducing a new action w m, of FN on TN, define d by (0 .4)  w m(S) = w ~S m ~ • A final remark pertains to the general construction in Section 1 . Ther e is no need to insist that N be f g to carry out the construction . Thu s Theorem 1 .1 may be extended to yield a function a . from the extende d genus of N to the extended genus of N (see [H31) . 1 . The constructio n Let N E 9't fs c 91 ; that is, N is a f g nilpotent group . There is the n a canonical exact sequenc e (1 .1) T N >--> N --» FN, TN = torsion subgroup of N , FN = torsionfree quotien t Now let a : TN -» i ' be a surjection, so that T is a finite nilpoten t group . Assume that ker a is normal in N ; call this condition K . The n we know [H1] that we may embed (1 .1) in a map of exact sequence s TN > 2 . -1N 7 -- r -* F N T ~ z N ~ FN 302  P . HILTO N with Ñ E J1 fg . Moreover, the LHS of (1 .2) is a push -out in the categor y of groups; and, obviously, FN = FN, TN = T indeed, we will ofte n write TN for T . We now replace N by a nilpotent group M in the genu s of N ; we will assume, as we may, that TM = TN and M p = Np fo r all primes p . We claim that ker a is normal in Munder the natura l embedding ker cx Ç TN = TM Ç M . For (ker a) p is normal in M p fo r all primes p, which shows that ker cx is normal in M . We thus have a commutative diagram TN ~ M — L » F M a  P ' TN  >--' F M Theorem 1 .1 . The association M H M defines a function a * v~( Ñ Proof : We have the commutative diagram (identifying F M p wit h FN p ) N ~  z p T   } N p  p } » F1V' p i p TNp ]  a p  m p  FM p p P 7r ' Now it is easy to prove that » F ~~N p ~ N p N p is also a push -out 1n the category of groups . Thus we have a (unique ) homomorphism : N p —> M p such that içfi p = f3p and k ip = i p . We MORPHISMS OF MISLIN GENERA  30 3 claim that rp ~ = * p . For * p O p _ rp, 3 p = 7r p = r p ~ p and p P . = ~pip = o =  Thus the diagra m ~ N ~ FN p P  P l R 2 p TN p >--> - ~ 1 ~ p  —» FM P M P commutes, showing that i is an isomorphism . This proves that  E 3 and establishes the theorem . ■ The following "functorial" properties of the association a 1—> a * ar e obvious . Theorem 1 .2 . (i) Id : TN ---~ TN satis fies the condition K an d Id . = I d (ii) lf a : TN ~ T = TN satisfies condition K and ~ :  T satis fies condition K, then & a satisfies condition K arid (d a ) * = & r * a * . Proof : (i) is trivial . As to (u), it suffices to remark that the existence o f 0 in (1 .2) guarantees that a satisfies condition K . Thus we superimpos e diagrams to produce TN >- N ~ F N a  Q TN > >- N ----~ F N TN >  >- N ~ F N and deduce, first, that c~a satisfies condition K and, second, that (da) * = ~* a * . For, just as (1 .3) was derived in similar manner to (1 .2) so 304 P . HILTO N TN ~~ N ~ F M F M TN >----> M ----~ F M is derived in a similar manner to (1 .4), and shows tha t = d,a .(M) (da) .(M) . • We now make the further hypothesis that N E No ; this is equivalen t to assuming that FN is commutative . Since FN = FN it follows tha t N E No, so that both C3 (N), 113 (N) are finite abelian groups . (Notic e that, in fact, N E No if and only if N E No .) We then hav e Theorem 1 .3 . Suppose that N E dto . Then a * : U(N) —> CU(N) is a homomorphism . Proof : Suppose thatK + L = M in O(N) . We continue to asum e that TK --=TL= TM =TN . Then, according to [HM], there exists an exhaustive pair (p : N --4 K , 1/) : N —) L, such that we may form the push -out (in % ) 1V ~ K T L ~ M We recall from [HM] that an exhaustive pair (9p,  is defined by th e requirement s (i) ~p or ~ is a T -equivalence, wher e T = T(N) = {p l N has p-torsion} ; and (u) for all primes p, (p or ~ is a p -equivalence . MORPHISMS OF MISLIN GENERA  30 5 However, examination of the proof of Theorem 2 .3 of [HM] show s that we may assume that both ~P and ~ are T-equivaZences . For havin g constructed cp as a T -equivalence, we defin e P= {po is not a p-equivalence } and then, modifying the argument in [HM], construct ~ to be a (Pu T ) - equivalence . With this strengthened sense of an exhaustive pair, we revert to (1 .6) . Then when restricted to TN, are both isomorphisms, so we ma y suppose that both are identities on TN . We may then suppose that a , r are also identities on TN . Now let us factor out ker a from each of K , L, M, N . Since ker a Ç TN, this gives rise to a commutative diagra m Ñ  `~  > K 7 P  T L  M which is easily seen to inherit from (1 .6) the property of being a pus h -ou t in 91 . Moreover, it is plain that , ~ remain T -equivalences and that , for all primes p, c,3 or ~ is a p -equivalence . Since TN is a quotient of T N it is plain that T(N) Ç T(N), so that and O are T(Ñ)-equivalence s and  is an exhaustiva pair . We conclude tha t k + L = M in O(N) , so that (p is a homomorphism . ■ 2 . A special cas e Since it has not yet provedpossible to calculate C3 ~N} systematicall y for N E No, it is not to be expected that we would have much succes s in trying to analyse the homomorphism a * in the generality in whic h it has been introduced in the preceding section . However, we do find i t possible to make some headway if we make the restrictiva assumptio n that N E N 1 . We then prov e Theorem 2 .1 . Let a * : C3 (N) —> ~ ~N} be defined as in Section 1 an d let N E 911 . Then a * is a surjective homomorphism . Proof : Since Ñ E 91o, it follows that N E No and a * is a homomorphism . Now C3 ~N} = 0 unless FN is cyclic [S], [HS] . Thus, to avoid 306  P . HILTO N triviality, we assume FN cyclic . Under this assumption, the top row o f (1 .2) splits for an action w : FN ~ Aut TN . Let o- : FN -4 N be a splitting (ii -a = 1), so that, if FN = (e), then w is given b y (2 .1)  w(e)(a) = yay –1 , a E TN, where y= o-(e) . We will often write e - a for w (O(a) . We use f3o- : FN ~ N to spli t the botton row of (1 .2) and write w : FN —> Aut TIV far the associate d action . Note that c~ is given b y j)(0(aa ) = a(w(e)(a)), a E TN . (2 .2 ) W e write (2 .2) more simply a s (2 .3)  a a= a(e -a ), aETN . Now let t be the height of ker D in FN ; that is, . since FN is cyclic, t is the arder of (D(O in Aut TN . Then, by the main theorem of [CH] , (2 .4)  03(N)  (Z/ .i)*/{+1} . Moreover, we may choose the isomorphism (2 .4) so that the grou p m prime to corresponding to [m] E (Z/)*/{±1}, is obtained from N simply by replacing the action D by a new action eJm , , defined b y (2 .5)  c~ m (e) (a) = cv (e m )(~), á E TN . Of course we have freedom in (2 .4) to choose rn within its given clas s [m] without changing 1 V m . We will, in fact, choose m to be a T'-number , where T = T(N) is the set of primes p such that N has p -torsion . T o see that we can do this it suffices to notice that m is prime to t so that , by Dirichlet's Theorem, the residue class [m] contains primes not in T . With such a choice of m, we show that N m, may be represented a s a * (Nm) for a suitable group N m in O(N) . We define Nm to be th e semi -direct product of TN and FN for the action wm : FN ~ Aut TN , given b y (2 .6)  wm (e) (a) = w(e m ) (a), a E TN . We first show that N m E O(N) . Consider the diagra m T N > > N,n ~ F N l m TN ~ N  >> FN MORPHISMS OF MISLIN GENERA  30 7 where the endomorphism of FN is just e f --> ~ m . Then (2 .6) asserts tha t (2 .7) satisfies the compatibility condition permitting us to complete i t with : N, n --> N to a commutative diagram . Now if p E T then m : FN —} FN is a p -equivalence, so that ~ : N m —> N is a p -equivalence . I f p T then TN p is the trivial group so both N and N,, L are pequivalen t to FN and hence p - equivalent to each other . Thus N, n E O(N) . Finally we show that a(N) - = N, . Consider the diagram s TN >~ N --» F N a 1  0 1 TN ~ N ---» F N TN ~ N, n --» F N a TN >  >- N, n -» F N Recall that we are writing "- " to indicate the actions of FN on TN o r TN in the first diagram ; let us write "o" for the actions of FN on TN o r TN in the second diagram of (2 .8) . Then (2 .3) ~ - aa = a (e • a), a E T N and (2 .6) e o a = em • a, a E TN . Moreover, by (2 .5), e o aa = e m • aa , a E TN . But since ~ . (y a = • a), it follows that e m • aa = = a(e m . a) , whence a( o a) = a(r . a) _ ~ m • aa =e o aa, a E TN . This, however, is precisely the compatibility condition guaranteeing th e existence, in the second diagram of (2 .8), of f3„ z : N, n  N„ n makin g the diagram commutative . Then  must be surj ective . This, however , guarantees that T N ~  13 m ? i n l N >---} .i Y r - n is a pus h -out in the category of groups and hence, by the uniqueness o f push-outs, that  = a * (N„, , ) . ■ We now consider the groups N , n E ~ ~N} constructed in the course o f our proof of Theorem 2 .1 . We have immediately 314  P . HILTO N [H3] HILTON P ., On the extended genus, Acta Math . Sinica 4(4 ) (1988), 372-382 . [HM] HILTON P . AND MISLIN G ., On the genus of anilpotent grou p with finite commutator subgroup, Math . Zeit . 146 (1976), 201-211 . [HS1] HILTON P . AND SCHUCK C ., On the structure of nilpoten t groups of a certain type, to appear . [HS2] HILTON P . AND SCHUCK C ., Calculating the genus of a certai n nilpotent group, Bulletin de la Sociedad Matemática Mexicana, t o appear . [M] MISLIN G ., Nilpotent groups with finite commutator subgroups , Springer Lecture Notes in Mathematics418, 1974, pp . 103-120 . [S] SCHUCK C ., Some contributions to the study of a class of nilpoten t groups, Ph . D . Dissertation, SUNY Binghamton, 1992 . Department of Mathematical Science s SUNY Binghamto n Binghamton, New York, 13902-600 0 U .S .A . Primera versid rebuda el 10 de Novembre de 1993 , darrera versis rebuda el 14 de Març de 1994