dedicated to h. wielandt on the occasion of his 90th birthday

On the Group of Autoprojectivities of aNonperiodic Modular Group

Dedicated to H. Wielandt on the occasion of his 90th birthday

M. COSTANTINI and G. ZACHER*Dipartimento di matematica pura ed applicata, Universita© di Padova, via Belzoni 7,35131 Padova (Italy). e-mail: {costantini, zacher}@math.unipd.it

(Received: 26 October 1999)

Communicated by K. Strambach

Abstract. We determine some relevant structural properties of the group of lattice auto-morphisms of a nonperiodic modular group.

Mathematics Subject Classi¢cations (2000). 20D30, 20Kxx, 06Cxx.

Key words. modular group, autoprojectivity

A given group G is called modular if its subgroup lattice L�G� is a modular lattice. In[1] and [2] we investigated the structural properties of the group P�G� ofautoprojectivities of a periodic modular group G. In the present paper, we willbe concerned with a similar problem in case G is a nonperiodic modular group.It is well known that in such a group, the set T of all periodic elements is a torsionsubgroup, while G=T is an Abelian group: actually G itself is Abelian as soon asthe rank of G=T is greater than 1 ([10]), in which case P�G� coincides with the groupPA�G� of linear autoprojectivities of G, i.e. of autoprojectivities induced by auto-morphisms ([1]). Our investigation is reduced to the nonperiodic modular groupsG with G=T (Abelian) of rank 1; since such groups are projective to Abelian groups([13]), we may assume, without loss of generality, that G is Abelian. In our situation,there exists a subgroup R of the additive groupQ of the rationals withZWR and anisomorphism Z of R onto G=T ([8]). Since the structure of P�R� has been determinedin [7], in what follows we shall always assume that:

�� G is an Abelian group with a nontrivial torsion subgroup T and with G=T ofrank 1.

*The authors are grateful to the MURST for the financial support during the preparation ofthis paper.

Geometriae Dedicata 85: 197^216, 2001. 197# 2001 Kluwer Academic Publishers. Printed in the Netherlands.

We shall refer to groups with the property �� simply as �-groups.The paper is divided into four sections. Section 1 contains, as preliminary, several

lemmata of general nature on projectivities, which shall ¢nd their application in thesuccessive paragraphs.

Given a �-group G with torsion subgroup T , the following two normal subgroupsof P�G� will play a relevant role in our investigation:

F �G=T � � fw 2 P�G� j wjG=T � 1g; F �T � � fw 2 P�G� j wjT � 1g:

It turns out that P�G� � F �G=T �F �T � (Theorem B).In our analysis, we are mainly interested in producing nonlinear autoprojectivities

and, more precisely, pointing out particular subgroups H of P�G� such thatP�G� � H � PA�G�, i.e. a description of P�G� modulo PA�G�. Since the structureof the automorphism group of an indecomposable �-group G is still not well under-stood, our most satisfactory results concerning P�G� are obtained under someadditional splitting assumption of G over certain Hall subgroups of T .

Section 2 deals with the group F �G=T �. Besides some general reduction steps givenin Lemmas 2.1 and 2.2, we show (Theorem A) that this group contains some relevantpro¢nite subgroups which are inverse limits of ¢nite groups of autoprojectivitiesalready studied in [3].

In Section 3, we analyse the group F �T �whose main structural properties are givenin (13) and Theorem C. Finally, in Section 4, Theorem D gives a fairly good descrip-tion of P�G�modulo PA�G�, in the case where G is a �-group satisfying some splittingcondition.

Our notation and terminology is standard, relying essentially on that used in [14].P is the set of all primes. For m=n 2 Q� with �m; n� � 1, we put

u�m=n� � fp 2 P j pjm or pjng:

C�G� is the semilattice of cyclic subgroups and PH �G� the group of autoprojectivities¢xing the subgroupH ofG. IfRW �Q;��,R� denotes the set of positive elements.Rn

is the local ring Z=pnZ. t�A� is the torsion subgroup of an Abelian group A.

1. Given a �-group G with torsion subgroup T , set p � p�T � � fpi 2 P j Tpi 6� 0g.Due to the existence of the isomorphism Z, it will be useful to recall some knownfacts concerning the group P�R� ([7, 14]). Given s 2 SymZ�, the permutationpi 7!pis of P extends in a unique way to an automorphism ~s of the semigroup�Q;��, by de¢ningY

i

pnii 7!Yi

pniis; 0 7!0:

~s induces an autoprojectivity of the group �Q;��, which is singular exactly at thoseprimes pi for which is 6� i. We have ZWR ~s and R ~s � R if and only if the type

198 M. COSTANTINI AND G. ZACHER

typ�R� of R is preserved by s, i.e. typ�R�s � typ�R�. For ; � o � P, we set

S�o� � fs 2 SymZ� j typ�R�s � typ�R� and is � i for i 2 og:

Moreover, if hpi �1� � mi is the pi-height of 1 in R and s 2 S�o�, we set

Ds � fi 2 Z� j mis 6� mig;as �

Yi2Ds

pÿmii 2 R �1 ifDs � ; �; qs � as=a ~s

s:

For every r 2 Q� we denote by r the automorphism x 7!xr of �Q;��. SinceR ~sqs � R ~s, the map cs � ~sqs induces an autoprojectivity cs of R and, sinceqst � q~t

sqt, one gets

(1) the position j: s 7!cs de¢nes a monomorphism of S�o� into P�R� such that

acss � as; cÿ1s rcs � r ~s; P�R� � �S�;��j j� PA�R�

and, for

P1�R� � fp 2 P j hp�1� � 1g;

PA�R� � hr 2 R� j u�r� � P1�R�i:

Given c in P�R�, then c is induced by the map sjq, where s 2 S�;� is uniquelydetermined by the singularities of c.

Given a �-group G, a j of P�G� has no singularities at the primes of p ([16], 3.1),hence the s relative to jjG=T belongs to S � S�p�. Notice that for g 2 G n T(2) the relation�zx � g is satis¢ed for an x 2 G and a z 2 Z if and only if the relation�z ~s ~x � ~g holds for h~gi � hgij, h ~xi � hxij.

We also recall that if the �-group G splits over T , i.e. G � R� T , then for a givens 2 S the map

�3� js � r� t 7!rcs � �rcs=r�t; for r 6� 0;t 7!t;

�with r 2 R, t 2 T , de¢nes an element of PR�G� such that jsjR � cs � ~sqs andjsjT � 1 ([12, 14], 2.6.16), while, by (1), the position s 7!js de¢nes a monomorph-ism of S into PR�G�. Finally, we remark that

(4) if G=T is cyclic, then js is independent of the complement R.

For suppose R0 is another complement of T , and let j0s be the corresponding map.We show that js � j0s. We have R � hri, R0 � hr0i, with r0 � er� t0, for certaine � �1, t0 2 T . Let x 2 G. If x 2 T , then xjs � x � xj0s. So assume x 62 T , hence

AUTOPROJECTIVITIES OF A NONPERODIC MODULAR GROUP 199

x � mr0 � t. Then

xjs � �mer�mt0 � t�js � �me�csr� �me�cs

me�mt0 � t� � mcser�mcs

m�mt0 � t�

� mcs�er� t0� �mcs

mt � mcsr0 �mcs

mt � �mr0 � t�j0s � xj0s;

and we are done.We are now going to collect a series of propositions useful in the study of P�G�,G a�-group.

Given a directed index set �L; W �, we call a family fUlgl2L of subgroups of agroup U a local system for U if U �Sl Ul and, for l, m 2 L with lW m we haveUl WUm. Clearly if S is any co¢nal subset* of L, then also fUZgZ2S is a local systemfor U .

LEMMA 1.1. Let fUlgl2L be a local system for a given group U and for each l 2 L letjl be a projectivity of Ul onto a subgroupU

jll of U, such that

Sl U

jll � U. Then there

exists a j 2 P�U� such that jjUl � jl for all l 2 L if and only if for l, m 2 L withlW m the relation

jmjUl � jl ��

holds. Moreover, j is uniquely determined by fjlg.If each jl, for l 2 L is a linear projectivity induced by a ¢nite number of

isomorphisms, then j 2 PA�U�.Proof. Uniqueness: let j, j0 be in P�U� with jjUl � j0jUl for each l 2 L. Now,

given X 2 C�U�, there exists a l 2 L such that X WUl; but then Xj0 �Xjl � Xj, so that j � j0.

Necessity: since Ul WUm one has

jl � jjUl � �jjUm�jUl � jmjUl:

Existence. Take an X 2 C�U� and let l 2 L be such that X WUl. De¢ne a map t onC�U� by setting t:X 7!X t � Xjl . Due to the condition ��, t is a well de¢nedinclusion preserving map. Since fUjl

l gl is also a local system of U , with the helpof fjÿ1l g we may de¢ne similarly a t0 on C�U�; but then clearly tt0 � t0t � 1, i.e.t 2 AutC�U�. Finally, suppose X , Y , Z in C�U� are such that X W hY ;Zi; then thereexists a l 2 L such that hY ;ZiWUl, so that

X t � Xjl W hY ;Zijl � hYjl ;Zjli � hY t;Zti

and conversely. Hence, t extends to a j 2 P�U� ([14], 1.3.2) and jjUl � jl.

*See [5] for a definition.


Finally, assume that for each l 2 L, jl is linear. Let j be the projectivity ofU suchthat jjUl � jl for each l. Since jl is induced by a ¢nite number of isomorphisms, by[14], 1.3.6, j is linear. &

Given a �-group G, let g, g0 be elements of G n T . Set C � C�hgi� n f0g,C0 � C�hg0i� n f0g and let ~C, ~C0 be co¢nal subsets of C, C0 by inverse inclusion,respectively. Consider two local systems fLX gX2 ~C , fL0X 0 gX 02 ~C0 for G such thathgiWLX , hg0iWLX 0 . With this notation in mind, we prove

LEMMA 1.2. Assume that there is given an isomorphism w of ~C onto ~C0 and a familyfjX gX2 ~C of maps with jX a projectivity of �LX=X � onto �LXw=Xw�. Then there exists aunique j 2 P�G� such that jj�LX=X � � jX if and only if

jX � jY j�LX=X � for every X ;Y 2 ~C with Y WX : ��

Proof. For j, c in P�G� and X in ~C, let jj�LX=X � � cj�LX=X �. If 0 6� D 2 C�G�,there exists an X0 2 ~C with DWLX0 , and X0 WD if D 6WT . We have

Dj �\X2 ~C

X WX0

�X �D�

0BBB@1CCCA

j

�\X2 ~C

X WX0

�X �D�j �\X2 ~C

X WX0

�X �D�jX

�\X2 ~C

X WX0

�X �D�cX �\X2 ~C

X WX0

�X �D�c � Dc

which proves the uniqueness. For the necessity of the condition, notice that

jX � jj�LX=X � � �jj�LY=Y ��j�LX=X � � jY j�LX=X �:To prove the suf¢ciency, one sees as a consequence of �� that jX preserves thep-indices and that for X WHWLX , H is in¢nite cyclic if and only if such isHjX , where HjX =X w � �H=X �jX . It follows that H � X � �H \ T � if and only ifHjX � XjX � �HjX \ T �. We are now going to de¢ne a t 2 AutC�G� realted tothe family fjX gX . For D 2 C�G�, there exists an X 2 C such that DWLX and alsoX WD if D 6WT . De¢ne

Dt � �D� X �jX \ T ; if DWT ;DjX ; if X WD:

(

We claim that t is a well de¢ned map. Assume, to begin, DWT , and let Y WX . Bysetting DX � �X �D�jX \ T , DY � �Y �D�jY \ T , using �� one gets

XjX �DX � �X �D�jX � �X �D�jY � �Y �D� X �jY

� �Y �D�jY � XjY � �YjY �DY � � XjX � XjX �DY ;


hence DX � DY , i.e. DX � Dt � DY . Assume now D 6WT ; from Y WX WD and�� we get XjX � X t � XjY .

Clearly the map t preserves inclusions and the same holds for the map t0 de¢ned onC�G� relative to the family fjÿ1X 0 gX 02 ~C0 . Since tt0 � t0t � 1, t lies in AutC�G�.

Let nowD, Y , Z be elements of C�G�withDWY � Z. There exists an X 2 C suchthat Y � ZWLX and with X contained in each of the groups D, Y , Z which is ofin¢nite order. To see that Dt WY t � Zt, we distinguish several cases.

�a1� Y � ZWTXjX �DX � �X �D�jX W �X � Y � Z�jX � �XjX � YX � � �XjX � ZX �, thus Dt �DX � �XjX �DX � \ T W �XjX � YX � ZX � \ T � YX � ZX � Y t � Zt.�a2� D� ZWT , Y 6WT .DWZ � Y implies DWZ, hence Dt WZt WZt � Y t.�a3� DWT , Y 6WT, Z 6WTX �DWY � Z implies XjX �DX WYjX � ZjX , hence Dt WY t � Zt.�a4� ZWT , D 6WT, Y 6WTSince DWY � Z � X � Z � Y we get Dt � DjX W �X � Z�jX � Y t � XjX � ZX�YjX � ZX � YjX � Zt � Y t.�a5� D 6WT , Y 6WT, Z 6WTDWY � Z implies Dt � DjX WYjX � ZjX � Y t � Zt.

Similarly, one concludes that DWY � Z implies Dtÿ1 WY tÿ1 � Ztÿ1 ; hence fromDt WY t � Zt follows DWY � Z. But now by [14], 1.3.2, t extends to aj 2 P�G�. From the de¢nition of t one then sees that for an X 2 C,jj�LX=X � � jX . &

Let G be a nonperiodic Abelian group and 0 6� Tp a primary component of itstorsion subgroup T . If one sets s2�Tp� � hOn�Tp� j On�Tp�=pOn�Tp� noncyclici, it isknown ([1, 14], 2.6.6) that

(5) if j 2 P�G� then jjs2�Tp� 2 PA�s2�Tp��.We have s2�Tp� 6� Tp if and only if Tp has the structure Tp � K � C, where K � Cpm ,1Wm 2N [ f1g, expC � ps, 0W s < m. In this case s2�Tp� � Cps � C. We noticethat T 0WTp implies s2�T 0�W s2�Tp�.

DEFINITION 1.1. We call an Abelian p-group of the formM � H � C, whereH �hai �hbi with 1 6� pn � j a jX j b j � pm X ps � expC, sX 0, mX 1 and s < m ifm � n a generalized �n;m; s�-group (if s 6� 0 an �n;m; s�-group, [3]) of ¢nite exponentpn, while A � �hai; hbi� is called the frame of H.


LEMMA 1.3. LetM � H � C be an �n;m; s�-group, C0 a subgroup of C of exponent ps

and M0 � H � C0. Let

~R�M0� � fr 2 P�M0� j Hr � H; rjC 0 � 1g;~R�M� � fr 2 P�M� j Hr � H; rjC � 1g:

Then c: ~R�M� ! ~R�M0�, r 7!rjM0 is an isomorphism.Proof. Let r be in ~R�M�. Since rjOs�M� 2 PA�Os�M��, rjOs�M� is induced by an

automorphism of the form �a; 1�, a 2 Aut�Os�H��; thus rjOs�M0� � �a; 1�. It followsthat if r 2 kerc, then a � 1 and rjH � 1, i.e. r � 1 by [2], 1.6: hence c is amonomorphism. Take now a w 2 ~R�M0�: there exists an �a; 1� 2 AutM0 such thatw0 � w�a; 1� 2 R�M0�. Now, as a consequence of [2], 5.5, one sees that w0 extendsto a ~w0 2 R�M�; but then ~w0�aÿ1; 1� � ~w 2 ~R�M� and ~wjM0 � w, i.e. c is alsosurjective. &

LEMMA 1.4. Let M � H � C be a generalized �n;m; s�-group with 0W sWm < n 2N, ~M � ~H � C a generalized �~n; ~m; s�-group with HW ~H, ~n 2N and where~m � s if m � s, ~A � �h~ai; h~bi� a frame of ~H, A � �hai; hbi� one of H with a 2 h~ai,b 2 h~bi. For mW t < n, set Nt � hptai and Mt �M=Nt � H � C. ChooseA � �hai � ha�Nti; hbi � hb�Nti� as a frame of H and let c be an element ofPA�Mt�* with cjhpmÿsb;Ci � 1. Then there exists a ~c 2 P ~A� ~M� such that

~cjhpmÿsb;Ci � 1 and ~cjMt � c.Proof. Due to Lemma 1.3, for the construction of ~c we may assume C ¢nite. Let

M� � Hom�M;Q=Z� be the group of characters of the Abelian group M; onM �M� consider the bilinear form given by hm; wi � w�m�. It de¢nes a duality dof M onto M�. Let �a; b; ci�i be a basis of H � C, �a�; b�; c�i �i the dual basis ofM� ([8], 13.2); then d induces a duality of �M=Nt� onto �Nt=0�, with

Ndt � hpnÿta�; b�; c�i jii � hpnÿta�; b�i � C�:

Now j� � dÿ1cd is an element of P�Ndt � ¢xing the frame �hpnÿta�i; hb�i� of

hpnÿta�; b�i; moreover j�jC� � 1. By [3], Theorem A, 1.7(a) and (18), j� extendsto a ~j� of PA��M��, A� � �ha�i; hb�i�. Set c � d ~j�dÿ1; then c 2 Pj�M� withCc � C, hence also hpmÿsb;Cic � hpmÿsb;Ci and cjMt � c. Moreover, for the samereasons quoted above, c extends to ~c 2 P ~A� ~M�. Since ~cjhNt; pmÿsb;Ci=Nt �cjhNt; pmÿsb;Ci=Nt � 1, we get ~cjhpmÿsb;Ci � 1. &

2. Given a �-group G and a nonempty subset x of p, set Tx �L

p2x Tp and de¢ne

F �G=Tx� � fr 2 P�G�j rj�G=Tx� � 1g; F �Tx� � fr 2 P�G�j rj�Tx=0� � 1g:

Then

*PA�Mt� the stabilizer of hai and h�bi in P�Mt�:


(6) F �G=Tx� and F �Tx� are normal subgroups of P�G�; moreover every element ofF �G=Tx� is strictly index preserving.

It is enough to show that j preserves the indices of the cyclic subgroups [15] and thisis certainly the case on the periodic subgroups ([16], 3.1). So let g be nonperiodic andp a prime; then

hgi=hpgi � hg;Ti=hpg;Ti � hg;Tij=hpg;Tij � hgij=hpgij:

Assume that for a certain x � p, G splits over Tx, i.e. G � A� Tx. Then we have thefollowing obvious factorization:

�7� F �G=Tx� � FA�G=Tx��F �Tx� \ F �G=Tx� \ PA�G��

Clearly j 2 FA�G=Tx� implies jjA � 1.To deal with FA�G=Tx�, we point out a useful reduction step. For p 2 x, set

Gp � A� Tp; thenG �P

p2x Gp and, for p1, p2 2 x and p1 6� p2,Gp1 \ Gp2 � A. UsingOre's criterion for intersection distributive pairs [11], one gets

(8) given any X WG, one has the decomposition X �Pp2x X \ Gp.

LEMMA 2.1. Given the �-group G of the form G � A� Tx �P

p2x Gp, x � p, forj 2 FA�G=Tx� set jp � jjGp. Then the map g:j 7!�jp�p2x is an isomorphism ofFA�G=Tx� onto F � Qp2x FA�Gp=Tp�.

Proof. Take j 2 FA�G=Tx�. By (8), for X WG we get Xj �Pp�X \ Gp�j �Pp�X \ Gp�jp , and one concludes that g is a monomorphism. Take now any�jp�p 2 F For X WG set

j:X 7!Xp

�X \ Gp�jp ; j0:X 7!Xp

�X \ Gp�jÿ1p :

Then j and j0 are inclusion preserving maps with jjA � j0jA � 1. Consider anX WG. Then X=X \ A is a x-group and one gets

Xj=�X \ A� �Xp

�X \ Gp�jp=�X \ A� �Mp

��X \ Gp�jp=�X \ A��;

i.e. �Xj=�X \ A��p � �X \ Gp�jp=�X \ A�. On the other hand, since Xj \ A � X \ Awe have �Xj=�X \ A��p � Xj \ Gp=�X \ A�. Thus Xj \ Gp � �X \ Gp�jp . SimilarlyXj0 \ Gp � �X \ Gp�jÿ1p . But now Xjj0 � �Pp�Xj \ Gp��jÿ1p �Pp��X \ Gp�jp�jÿ1p �P

p�X \ Gp� � X . Similarly, one sees that Xj0j � X . Therefore j is bijective andsince it preserves inclusions in both directions, j 2 P�G�. Moreover, jjGp � jp, thatis j 2 FA�G=Tx� and jg � �jp�p. &

In case that G splits over a Hall subgroup Tx of T , the investigation of F �G=Tx� isso, by Lemma 2.1, reduced, modulo F �Tx� \ PA�G�, to that of FA�Gp=Tp�, p 2 x.So, without loss of generality, we shall assume x � fpg.


Fix an a0 2 A n t�A�, and 0 6� n 2N. Set Cn � C�hna0i� n f0g; then Cn is, by inverseinclusion, a directed set. Let fLX gX2Cn

denote a local system of G � A� Tp such thatLX � AX � TX with ha0iWAX WA, TX WTp so, for X , Y in Cn, we have AX WAY ,TX WTY as soon as Y WX . By setting LX � LX=X � AX=X � �TX � X �=X �AX � TX , we have TX � TX and AX � PX � P0X is a periodic group where thep-primary component PX is quasi-cyclic or cyclic, depending upon whether thep-height of a0 � T in G=T is in¢nite or ¢nite; we rewrite LX � P

0X �MX , with

MX � PX � TX � �AX \MX � � TX the p-primary component of LX . In our inves-tigation we are going to take advantage of the results obtained in [2] and [3]. Inthis context we shall, moreover, assume that the given local family fLX g has thefollowing properties:

Case 1. If expTp <1, or expTp � 1 and s2�Tp� 6� Tp, then we have Tp � K � C,with expK � expTp, and K cyclic or quasicyclic. Then we assume LX �AX � TX , TX � KX � C, with KX � K \ TX .

Case 2. If expTp � 1 and s2�Tp� � Tp, then we choose LX such that s2�TX � � TX .It is clear that such local systems exist (for instance we may take LX � G for every

X ). We now introduce

~FA�G=Tp� � fr 2 P�G� j rjA � 1; rjs2�Tp� � 1 and rjK � 1 if s2�Tp� 6� Tpg:

For a given j 2 ~FA�G=Tp� we get

jX � jjLX � �j0X � jjP0X � 1;j00X � jjMX �;

so jjLX is completely determined by jjMX . Set

S�MX � � fr 2 P�MX � j rjPX � 1; rjs2�TX � � 1 and rjKX � 1 if s2�TX � 6� TX g:

Taking into account (5) one may derive the following properties:

(9) �a� F �G=Tp� � ~FA�G=Tp��PA�G� \ F �G=Tp��b� for X, Y 2 Cn and Y WX , pYX :S�MY � ! P�MX �, r 7!rjMX is a homo-

morphism into S�MX �,�c� ~FA�G=Tp�XFA�G=Tp� \ F �Tp� and equality holds if and only if s2�Tp� � Tp,�d� if j 2 ~FA�G=Tp�, then j00X � jjMX 2 S�MX � and j00X � jY jMX if Y WX.

Notice that given in a �-group an element a 2 G n T and a j 2 F �G=T �, there existsan a0 2 hai, a0 6� 0, such thatjjha0i � 1. In fact we have haij � ha� ti for some t 2 Tso, by (6), ha0 � j t jaij � j t jhaij � j t jha� ti � ha0i. Now, for T WHW ha0;Ti, wehave Hj � H, hence �ha0i \H�j � ha0i \H, that is jjha0i � 1.

LEMMA 2.2. Given the �-group G, assume that G splits over Tp: G � A� Tp, and letC be a co¢nal subset of Cn. Then �a� given a family fj00X 2 S�MX �gX2C, there exists oneand only one j 2 ~FA�G=Tp� such that for X 2 C jX � jjLX � �j0X � jjP0X � 1;


j00X � jjMX � if and only if for X, Y in C

Y WX implies j00X � j00YjMX:

�b� The restriction map r:FA�G=T � ! P�A�, j! jjA is an epimorphism ontoF �A=t�A��, and FA�G=T � splits over ker r � FA�G=Tp�.

Proof. �a� if one applies Lemma 1.2 to C1 � C, w � 1 and to the familyfjX � �j0X � 1 on P

0X ;j

00X on MX�gX2C of autoprojectivities of f�LX=X �gX , the con-

clusion follows.�b� given any d 2 F �A=t�A�� there exists, as noted before, an a0 2 A n t�A� such that

djha0i � 1. De¢ne, for X 2 Cn, jX 2 P�LX � by jX � �j0X � djP0X ;j00X � 1 on MX�.Since for Y WX , jX � jY jLX , by Lemma 1.2 there exists a unique jd 2 P�G� suchthat jdjLX � jX , i.e. jdj�AX=X � � dj�AX=X � and jdjMX � 1. Hence jdjA � dand jdjTp � 1, in particular jd 2 FA�G=T � and since r is a homomorphism intoF �A=t�A��, we conclude that r is actually an epimorphism onto F �A=t�A��. Moreover,j 2 ker r if and only if jjA � 1, that is ker r � FA�G=Tp�. The mapj:F �A=t�A�� ! FA�G=T �, d 7!jd, is a monomorphism, hence FA�G=T � splits overker r with �F �A=t�A��j as a complement. &

In studying ~FA�G=Tp�, it will be convenient to separate the case expTp � 1 ands2�Tp� � Tp from the remaining cases. We deal ¢rst with the situation

Case 1. expT is either ¢nite, or in¢nite with s2�T � 6� T .One has Tp � K � C with K � Cpm , 0 < m 2N [ f1g, expK � expTp, expC � ps

¢nite with 0W s < m if s2�Tp� 6� Tp. In order to use the structure theory developedin [2] and [3] for R�MX �, MX a generalized �n;m; s�-group of ¢nite exponent, weshall make reference to a convenient local system fLX gX2Cn

of G. Starting withthe trivial local system fLX gX where, for every X , LX � G, LX � P

0X � PX � Tp,

we construct a new local system which we still call fLX gwith the following procedure:�a1� A=t�A� � G=T � R is not p-divisible.For X 2 C1 � C�ha0i� n f0g, let pnX be the p-index of X in A; set LX �LX=X � P0X � OnX �PX � Tp�; then fLX � AX � TX gX2C1

is a local system witha0 2 AX and the p-component MX � PX � TX of LX has exponent pnX . Choose now

n � pm�1 if expT � pm; finiteps�1 if expT � 1 :

�Then forX 2 Cn, LX � P

0X �MX withMX � PX � TX � PX � KX � C � HX � C,

where KX � LX \ K , KX � �KX � X �=X , and MX is a generalized �nX ;mX ; s�-groupwith nX > mX � mX sX 0 if expTp � pm, m <1, while MX is a generalized�nX ; s�-group if expTp � 1.

�a2� R is p-divisible.For X 2 C1, let pnX be the p-index of X in ha0i. With the same procedure used in a1�,de¢ne LX � LX=X � P

0X � O2nX �PX � Tp�; then �LX � AX � TX �X2C1

is a local sys-


tem of G with a0 2 AX . By choosing n as before, one sees that for X 2 Cn, ifexpTp � pm is ¢nite, MX is a a generalized �2nX ;mX ; s�-group with nX > mX �mX sX 0, if expT � 1, MX is a generalized �2nX ; s�-group.

In conclusion, for a given j 2 ~FA�G=Tp�, we have jjLX � �j0X � 1;j00X � jjMX �,where j00X 2 S�MX � and, for Y WX , j00X � jY jMX with j00X ¢xing the frameAX � �PX ;KX � of HX , being KX � K \ LX . If s � 0, S�MX � appears as the groupPAX�HX �, which has been described in []; if s 6� 0, an element j00X of S�MX � is related

to an element cX of RAX�MX � by means of a convenient elation �1; lX ; lX � where

lX 2 U�RmX �. Thus if one considers the group LX � f�1; l; l� j l 2 U�RmX �g �U�RmX �, one has

(10) RAX�MX � /S�MX � � RAX

�MX �LX ; RAX�MX � \ LX � ker r,

where r:U�RmX � ! U�Rs� is the natural projection.

LEMMA 2.3. For X, Y in Cn and Y WX, the restriction map pYX :S�MY � ! P�MX �,r 7!rjMX is an epimorphism onto S�MX �.

Proof. Let pt be the p-index of Y in X . If t � 0, we have MX �MY , and we aredone. Thus assume t > 0 and, without loss of generality, Y � ptX . Take a c inS�MX �. By Lemmas 1.3 and 1.4 there exists a ~c 2 S�MY � such that ~cjMX � c.&

THEOREM A. Let G be a �-group and, for a p 2 p, assume that G � A� Tp withs2�Tp� 6� Tp if expTp � 1. Then there exists a positive integer n such that forany X 2 Cn � C�hna0i� n f0g, MX is a generalized �nX ;mX ; s�-group. Moreover,

�a� for X, Y, Z in Cn and ZWY, �S�MX �; pZY � is an inverse system*

�b� the map j:j 7!fj00X gX2Cn0, �jX � jjLX � �j0X � 1;j00X � jjMX � is an isomorph-

ism of ~FA�G=Tp� onto the inverse limitlim ÿX2Cn

S�MX �.�c� for any X 2 Cn, the restriction map rX : ~FA�G=Tp� ! S�MX �, w 7!wjMX is an

epimorphism.

Proof. The existence of n has already been discussed, and statement �a� followsfrom (9) �b�.�b�: take a j 2 ~FA�G=Tp� and for X 2 Cn set, as usual, jX � �j0X � 1;j00X jMX �.

Then for ZWY , j00Y � j00ZjMY , i.e. fj00X gX2Cn2 lim ÿ

X2CnS�MX �, being j00X 2 S�MX �

by (9)d. Conversely let fj00X gX2Cn2 lim ÿ

X2CnS�MX �. Then by 2.2 there exists one (and

only one) j 2 ~FA�G=Tp� such that jjMX � j00X , i.e. jj � fj00X gX2Cn

, and the con-clusion follows.�c� is a consequence of Lemma 2.3. &

COROLLARY 2.4. The group ~FA�G=Tp� is a residually ¢nite group and evenresidually solvable as soon as Tp is either not locally cyclic or p < 5.

*see [4] for a definition of inverse system.


Proof. If c 2TCnker rX , then c ¢xes every cyclic subgroup of G, hence c � 1.

Moreover, S�MX � is, by [2] and (10), a ¢nite group and, by [3] even solvable sinceif Tp is not locally cyclic, MX is an �nX ;mX ; s�-group with s 6� 0. In case Tp is locallycyclic, from [9] one deduces that S�MX � is solvable if and only if p < 5. &

Before we proceed to examine the case 2, i.e. expTp � 1 and s2�Tp� � Tp, let usconsider in more details the particular situation in which expTp � pm is ¢niteand s2�Tp� � Tp, so that Tp � hbi� C, with j b j � pm � expC � ps. Here by (9) (c),we know that ~FA�G=Tp� � FA�G=Tp� \ F �Tp�:� L. Our considerations in [3], Section1, now allow us to get a more explicit description of the group L.

�a1� A=t�A� is not p-divisible.Set n � ps�1; then for X 2 Cn,MX is an �nX ; s; s�-group, where nX is the p-index of Xin A. We know by Theorem A, that a j 2 L is completely determined by the familyfj00X g, where j00X � jjMX 2 S�MX �. There exists a unique l0 2 U�Rs� such thatcX � j00X �1; lÿ10 ; lÿ10 � 2 RAX

�MX �, where AX � �PX ;KX �: actually if one setsL � f�1; l; l� j l 2 U�Rs�g � U�Rs�, then S�MX � � RAX

�MX � j�L. Take anyY 2 C�G� n C�T �, so Y � ha� ti with a 2 A, t 2 Tp, and let pk be the p-index ofhai in A. Without loss of generality we may assume X < Y and that0W k < nX ÿ s. Thus, by [3], ha� ticX � ha� m0kti, with m0k � 1 psÿ1Rs, henceha� tij00X � ha� mkti, where mk � m0kl0 and for 0W k1; k2 < nX ÿ s, we getmk1 � mk2 psÿ1Rs. In conclusion, given a j in L, then for c � jjC�G� 2AutC�G�, we have cjC�T � � 1 and cjC�G� n C�T � is de¢ned by ha� ti 7!ha� mkti where mk 2 U�Rs� depends only on the p-index of hai in A, while forki 2N, mk1 � mk2 psÿ1Rs.

Conversely, suppose we are given a sequence fmkgk2N in U�Rs� satisfying the con-dition mk1 � mk2 psÿ1Rs. We claim that c: ha� ti 7!ha� mkti lifts to an auto-projectivity j in L. Note that c preserves inclusion, and that if we de¢ne c0

starting from fmÿ1k g, we get cc0 � c0c � 1, so that c 2 AutC�G�. Takehyii 2 C�G� and assume that hy1iW hy2; y3i. We show that hy1ic W hy2ic � hy3ic.There exists an X 2 Cn such that hy2; y3iWLX and with X contained in each ofthe groups hyii which is of in¢nite order. Note that Xc � X . SupposeMX � < a > �Tp, a � a� X , a 2 A with p-index pk in A. By [3], theorem 1.5, thereexists a unique autoprojectivity ~c ofMX such that hpia� ti ~c � hpia� mk�iti (we notehere that formulas �c� and �c0� in [3], before Theorem 1.5, should read mnÿsÿk insteadof mk). Extend ~c to an autoprojectivity ~c of LX by de¢ning ~cjP0X � 1. We distinguishseveral cases:

�a01� hy2; y3iWT .

Then hyiic � hyii for every i, and we are done.

�a02� hy1; y3iWT , hy2i 6WT .

Then hy1iW hy3i, so that hy1ic W hy3ic W hy2ic � hy3ic.


�a03� hy1iWT , hy2i 6WT , hy3i 6WT .

Then �hy1i � X=X � ~c W �hy2i=X � ~c � �hy3i=X � ~c, so that hy1ic W hy2ic � hy3ic.�a04� hy3iWT , hy1i 6WT , hy2i 6WT .

Then �hy1i=X � ~c W �hy2i=X � ~c � �hy3i � X=X � ~c, so that hy1ic W hy2ic � hy3ic � X .But X W hy2ic, hence hy1ic W hy2ic � hy3ic.�a05� hy1i 6WT , hy2i 6WT , hy3i 6WT .

Then �hy1i=X � ~c W �hy2i=X � ~c � �hy3i=X � ~c, so that again hy1ic W hy2ic � hy3ic, andwe are done.

Since an analogous conclusion holds for c0, one concludes that c lifts to anautoprojectivity of G, which actually belongs to L. We note that the elements of~FA�G=Tp� \ PA�G� correspond to the families fmkgk such that mk1 � mk2 , hence~FA�G=Tp� \ PA�G� is isomorphic to L. Denoting by K�G� the group of auto-projectivities corresponding to the families fmkg with m0 � 1, we get

(11) ~FA�G=Tp� \ F �Tp� � K�G�� ~FA�G=Tp� \ PA�G�� with K�G� \ PA�G� � 1 andK�G� �Qi2N Hi, being Hi � Cpÿ1 if s � 1, Hi � Cp if sX 2.

�a2� A=t�A� is p-divisible.

Consider an ascending chain ha0; t�A�i � A0 < A1 < � � � < Ak < � � �, of subgroups ofA such that Ak=t�Ak� is not p-divisible and with [Ak � A. Set Gk � Ak � Tp andlet pnk be the p-index of A0 in Ak. Using �a1� one sees that ~FAk�Gk=Tp� is isomorphicto the subgroup �Qÿnk W i<�1Hi�L of �Qÿ1<i<�1Hi�L. Since G � [Gk one con-cludes that

�12� ~FR�G=T � � lim ÿ

k

Yÿnk W i<�1

Hi

0@ 1AL � Yÿ1<i<�1

Hi

!L:

After this digression, let us deal with

Case 2. expTp � 1 and s2�Tp� � Tp.ForX 2 C1, the groupMX as de¢ned in Case 1, is a group of ¢nite exponent pn

0X*, for

which s3�MX �:� hOi�MX � j jOi�MX �=pOi�MX � jX p3i �MX . Hence by [1, 14], wehave S�MX �WPAAX

�MX �; more precisely, since for r 2 S�MX �, rjTX � 1, r isinduced by an automorphism of the form �1; lX ; lX �, lX 2 U�RmX �. SincePotTp � Up, the multiplicative group of p-adic units, one concludes thatS�MX � � U�RmX �; in particular, for Z, Y in C1 with ZWY , the restriction mappZY :S�MZ� ! S�MY � is an epimorphism.

*n0X stands for nX in case �a1�, for 2nX in case �a2�.


THEOREM 2.5. Let G be a �-group and assume that, for a p 2 p, G splits over Tp.Then F �G=Tp�WPA�G� if and only if expTp � 1 and s2�Tp� � Tp.

Proof.Necessity. Assume expTp ¢nite or expTp � 1 but s2�Tp� 6� Tp. Then, from[3], we know that for most X W hna0i we have S�MX � 6WPA�MX �. But now, bytheorem A, there exists a j 2 ~FA�G=Tp� such that jjMX � j00X 62 PA�MX �. Hencej 62 PA�G�.

Suf¢ciency. By (9)(a), it is enough to show that ~FA�G=Tp�WPA�G�. From what wehave seen in Case 2, for X 2 C1 and j 2 ~FA�G=Tp�, j00X � jjMX is induced by anautomorphism of the form �1; lX ; lX �, lX 2 U�RmX �. Since for Y WX ,�1; lY ; lY �jMX � �1; lX ; lX �, we get lim ÿ lX � l 2 Up. Let c 2 PA�G� be inducedby 1� l. Then c 2 ~FA�G=Tp� and cjMX � jjMX ; so by Lemma 2.2 we getc � j and we are done. &

3. Given a �-group G, assume that G splits over Tx, x a convenient nonempty subsetof p�T �: G � A� Tx. From (7), Lemma 2.1, (9), (10), Theorem A, (11), (12) andCorollary 2.4 we get a fairly good structural description of FA�G=Tx� moduloPA�G�. We bring now our attention to the group F �T �.

LEMMA 3.1. Let G be a �-group which splits over T, i.e. G � R� T, whereZWRWQ. Then there exists a monomorphism j:S � S�p� ! FR�T �, s 7!js suchthat

jsjR � ~sqs;FR�T � � ��FR�G=T � \ F �T �� j�Sj��FR�T � \ PA�G��:

Proof. By (3), j: s 7!js de¢nes a monomorphism of S into FR�T � and by (1)jsjR � cs � ~sqs. Given now a j 2 FR�T �, there exists a unique s 2 S, determinedby the singularities of jjR, such that jÿ1s jjR 2 PA�R�; hence there exists am 2 PA�G� \ FR�T � such that

mÿ1jÿ1s j � r 2 F �G=T � \ FR�T � /FR�T � /FR�G�;

therefore j � jsmr � jsr0m; moreover Sj \ �FR�G=T �PA�G�� 1. &

Note that a m of FR�T � \ PA�G� is induced by an automorphism of the forma�Qp lp, a 2 PA�R� � hr 2 R� j u�r� � P1�R�i, and lp 2 Lp � AutCpm ifexpTp � pm, m <1, lp 2 Lp � Up if expTp � 1. Therefore

(13) FR�T � \ PA�G� � PA�R� � �Qp2p Lp�.Let G be a �-group and m a ¢xed isomorphism of R onto G=T ; for r 2 R�, choose a

representative gr in rm, hence rm � gr � T . Clearly fLgr � hgr;Tigr2R� and fhr;Tigr2R�are local systems, respectively, of G and ~G � R� T . Take a s 2 S � S�p� and, using(3), pick the corresponding js in FR� ~G�. Consider an element r 2 R� such thathaiW hri, where a stands for the positive rational number as de¢ned in Section


1. Then a � nr for a unique n 2 Z�. Assume now u�n� � p; then given theisomorphism Wr: hr;Ti ! hgr;Ti, mr� t 7!mgr � t,

(14) the map ws;r � Wÿ1r jsWr is an autoprojectivity of Lgr

since hrijs � h�1=n�aijs � h�1=n� ~sai � h�1=n�ai � hri and, by (4), ws;r does not dependon the choice of the representative gr in Lgr . Moreover one also checks that

(15) for hriX hr0iX hai, ws;rjLgr0 � ws;r0, hgriws;r � hgri and, for s, t in S, wst;a � ws;awt;a.

Take a g 2 G n T and assume that for X 2 C � C�hgai� n f0g, the index n �jhgi: hgi \ X j is a p0-number. If hg0i � �hgi \ X �ws;a , then by (2) there exists a unique~g 2 G n T such that

(16) g0 � n ~s ~g, while hmgi 7!hm ~s ~gi extends ws;ajhgi \ X to a unique projectivity cs;X ofhgi onto h~gi and, for s, t 2 S, cst;X � cs;Xct;X .

Let �m1;m2; . . .� be the height-vector of hga � Ti in G=T . If for X 2 C,

nX � jhgai:X j �Yi

pni;Xi ;

de¢ne

`i;X � ni;X ; if ni;X ; Wmi;

mi; if ni;X > mi;fX �

Yi

p`i;Xi :

(

Given X 2 C there exists in �G=T � a unique rX 2 R� such that jLgrX:Lga j � fX ; fLgrX

gis a local system of G and, forY WX , LgrY

XLgrX. We have the direct decomposition

LgrX=X � L�p�grX =X � L�p

0�grX=X ;

respectively in its p and in its p0-components. Notice that L�p0�

grX� hgX i is cyclic and

L�p�grX XT . Let r 2 R� be such that Lgr � hL�p�grX ; gai; then by (14), on Lgr there is de¢nedthe autoprojectivity ws;r where, by (15), ws;rjLga � ws;a. Since jhgX i:X j is a p0-number,there exists by (16) a unique projectivity cs;X of hgX i onto h~gX iXX ws;a which extendsws;ajX :X ! Xws;a . For the given X 2 C and s 2 S, we are now in the position tode¢ne a projectivity js;X of the interval �LgrX

=X � onto �LgrXws;a=X ws;a � by setting

(17) js;X � �j0s;X ;j00s;X �, wherej0s;X � cs;X j�hgX i=X �: �hgX i=X � ! �h~gX i=X ws;a �,j00s;X � ws;rj�L�p�grX =X �: �L

�p�grX=X � ! �L�p�grXws;a

=X ws;a �.

We have, for Y WX

(18) js;Y j�LgrX=X � � js;X .

THEOREM 3.2. [7] Let G be a �-group. Then there exists a monomorphism j: s 7!js

of S � S�p� into F �T �.


Proof. Take a s 2 S and, for X 2 C � C�hgai� n f0g, consider the projectivityde¢ned in (17). With reference to the automorphism w � ws;ajC and to the familyfjs;X gX of projectivities, 1.2 applies since (18) holds. Therefore there exists a uniquejs 2 P�G� such that jsj�LgrX

=X � � js;X : thus js 2 F �T �. Finally with the help of (15)and (16) one concludes that j is a monomorphism. &

Remark. Assume G � A� Tx for a certain x � p. Then Sj leaves A stable. In fact,since A=t�A� � G=T , we may choose each representative gr in A. But then, givens 2 S, our costruction of ss;X shows that gX and ~gX are in A. Hence�A \ LgrX

�js;X WA, and we are done.

THEOREM B. Let G be a �-group. Then P�G� is the product of its two normal sub-groups F �G=T � and F �T �.

Proof. Take a j 2 P�G�; then jjG=T determines a unique s 2 S�p� in correspon-dence to the p-singularities of j on R. By Theorem 3.2, we may considerjs � sj 2 F �T �; then jÿ1s jjG=T 2 PA�R�, hence it is induced by the automorphismx 7!xq, q a convenient element of R (see (1)). Set x � p \ u�q�. If x � ;, x 7!xqde¢nes a w 2 PA�G� \ F �T �; if x 6� ;, [7], proposition 4, tells us that G splits overTx: G � A� Tx. Then the map

g:x 7!xq; if x 2 A;x 7!x; if x 2 Tx

�

is an automorphism of G inducing a w of F �T � \ PA�G�, and we havejÿ1s jjG=T � wjG=T , i.e. r � wÿ1jÿ1s j 2 F �G=T �. Thereforej � jswr 2 F �T ��F �T � \ PA�G��F �G=T � � F �T �F �G=T �. &

We observe that our proof shows that

(19) Given a �-group G and a j in F �T �, there exists a unique s 2 S�p�, a w inF �T � \ PA�G� /PA�G� and a r 2 F �G=T � \ F �T � /P�G� such thatj � jsrw 2 �Sj j� n�F �G=T � \ F �T ��F �T � \ PA�G��.

Combining (19) with Theorem B one gets

(20) P�G� � �F �G=T � j�Sj��F �T � \ PA�G��;

i.e. P�G�, modulo PA�G�, is completely determined by the group of autoprojectivitiesF �G=T � j�Sj . In case that the �-group G splits over Tp, G � A� Tp, one may even saythat P�G�, modulo PA�G�, is determined by FA�G=T � j�Sj . According to Lemma2.2(b);

(21) FA�G=T � � F �A=t�A��jFA�G=Tp� � F �A=t�A��j ~FA�G=Tp��PA�G� \ FA�G=Tp��,~FA�G=Tp� being described by Theorem A.


Moreover, one has

FA�G=Tp� \ F �T �WFA�G=Tp� \ F �Tp�W ~FA�G=Tp�WFA�G=Tp�:

Taking into account (11), (12) and Theorem 2.5, to determine F �T � \ FA�G=Tp�inside the inverse limit

lim ÿX2Cn

S�MX � (see Theorem A), we are left to investigate

the situation in which s2�Tp� 6� Tp.For a j 2 FA�G=Tp� \ F �T � and an X 2 Cn, one has jX � jjLX � �1;j00X �, with

j00X 2 Q�MX � � fr 2 S�MX � j rjTX � 1g. Since for Y WX , TX WTY , the restrictionmap pYX :Q�MY � ! P�MX � is a homomorphism into Q�MX �. Hence

(22) �Q�MX �; pZY � is an inverse system, with Q�MX � � RA�MX �U�RmX � .RA�MX �,where A � �PX ;KX �. In particular for s > 0, Q�MX � is a ¢nite soluble group [3].

PROPOSITION 3.3. Given a �-group G of the form G � A� Tp with s2�Tp� 6� Tp,

consider the isomorphism j: ~FA�G=Tp� ! lim ÿX2Cn

S�MX �; then the restriction map

j0 � jjFA�G=Tp� \ F �T � is an isomorphism ontolim ÿX2Cn

Q�MX �.

Proof. Given fj00X g 2lim ÿX2Cn

Q�MX �, let j 2 ~FA�G=Tp� be such that jj � fj00X g Sincej00X jTX � 1, where TX � TX WT , and [TX � T , one concludes that jjT � 1, i.e.j 2 ~FA�G=Tp� \ F �T � � FA�G=Tp� \ F �T �, and the conclusion follows. &

Recall that according to [3], G�M� is the subgroup of R�M� consisting of elementsfor which �m� � �1�. We are going to prove

LEMMA 3.4. Let M � H � C be an �n;m; s�-group with nXm > s > 0 andA � �hai; hbi� a frame of H. Then RA�M� \Q�M� � GA�M�.

Proof. Let j:R�M� ! Fn;m;s be the isomorphism as de¢ned in [2], 5.5, with ref-erence to the frame A, and let c 2 RA�M� \Q�M�. Then cjOs�1�M�=O1�H� ��1; m; m� � �1; 1; l�, where l, m 2 U�Rs�. It follows that m � 1, henceRA�M� \Q�M�WGA�M�. Conversely let �s0; s1; . . . ; snÿm; tnÿm� 2 GA�M�j. Onehas htiW hbi� C if and only if hti � hpk�ia� b� � ci with i � 0; but now forc � �s0; s1; . . . ; snÿm; tnÿm�jÿ1 , htic � hpk�0tnÿma� b� � ci � hpkb� ci � hti, hencecjhb;Ci � 1, so GA�M�WRA�M� \Q�M�. &

LEMMA 3.5. Under the assumption of 3.3, pYX :Q�MY � ! Q�MX �, r 7!rjMX is anepimorphism.

Proof. �a� s > 0.Write

MX � PX � KX � C; MY � PY � KY � C;


where KX WKY . SetM � PY � KX � C and take a c 2 Q�MX �; by Lemmas 1.3 and1.4, there exists a c 2 PPY ;KX

�M� such that cjMX � c; butcjKX � C �cjKX � C � 1, so c 2 Q�M�. Now there exists a lY 2 U�RnY � such thatr � c�lY ; 1; 1� 2 Q�M� \ RPY ;KX

�M� � GPY ;KX�M� by Lemma 3.4. With the help

of [3], Theorem A, one produces a ~r 2 GPY ;KY�MY � such that ~rjM � r. By 3.4,

~r 2 Q�MY �, hence also j � ~r�lÿ1Y ; 1; 1� 2 Q�MY �. But now ~r�lÿ1Y ; 1; 1�jM �r�lÿ1Y ; 1; 1� � c, and since cjMX � c, we have that for j 2 Q�MY �,jjMX � c 2 Q�MX �.�b� s � 0.

We have MX � PX � KX and Q�MX � � PAX�MX �. Using 1.4 one again reaches the

conclusion. &

With the notation so far introduced, we are going to prove

THEOREM C. Given a �-group G, assume that for a p 2 p, G splits over Tp,G � A� Tp, that s2�Tp� 6� Tp and that Tp is not locally cyclic. Then there exists afamily of normal subgroups fNX gX2Cn

of L:� FA�G=Tp� \ F �T � such thatTX NX � 1 and for X 2 Cn, L=NX � Q�MX �.Proof. By Proposition 3.3, L � lim ÿ

X2CnQ�MX � and, by Lemma 3.5,

rX :FA�G=Tp� \ F �T � ! P�MX �, r 7!rjMX is an epimorphism onto the ¢nite soluble

group Q�MX �. Set NX � ker rX ; if c 2 \XNX , then cj�LX=X � � 1 which implies

c � 1, being fLX gX2Cna local system of G. &

We remark that if G is a �-group which splits over T , then by Lemmas 2.1, 3.1 �b�,Theorem C and along with (11), (12), (13) and (19), one gets a quite satisfactorydescription of P�G� modulo PA�G�.

4. Given the �-group G, since our interest lies in the determination of non-linearautoprojectivities of G, we may restrict our considerations (due to formula (20))to the group F �G=T �, even assuming that G does not split over T . Recall the localsystem D � fLgr � hgr;Ti j r 2 R�; W g of G introduced in the previous section.

For d 2 D, by what has been said in the previous sections, the structure of P�d�modulo PA�d� is fairly well understood. For j 2 F �G=T � and d 2 D we setjjd � jd 2 F �d=T �: by (6) we know that jd preserves the heights of the elementsin G; it follows that the map rd :F �G=T � ! F �d=T �, j 7!jd is a homomorphisminto Fh�d=T � � fw 2 F �d=T � j w preserves the heights in Gg. Now one sees thatthe families �F �d=T �; pd2d1 �d2D, �Fh�d=T �; pd2d1 �d2D, �F �d=T � \ PA�d�; p

d2d1�d2D are inverse

systems. The following holds

PROPOSITION 4.1.

�a� lim ÿd

F �d=T � � lim ÿd

Fh�d=T �


�b� j:F �G=T � ! lim ÿd

F �d=T �, w 7!fwdgd,�c� j:F �G=T � \ PA�G� ! lim ÿ

d�F �d=T � \ PA�d��, w 7!fwdgd

are isomorphisms.

Proof. �a� Let fwdgd 2lim ÿd

F �d=T � (fwdgd 2lim ÿd�F �d=T � \ PA�d��) (serve?), then

wd preserves the heights in d by (6) and since for d 0X d, wd � wd 0 jd, wd 2 Fh�d=T �, i.e.fwdgd 2

lim ÿd

Fh�d=T �.�b�, �c� The map j is clearly a monomorphism. Now given fwdgd 2

lim ÿd

F �d=T �(fwdgd 2

lim ÿd�F �d=T � \ PA�d��), taking into account that Pot d � f�1g, one sees by

using Lemma 1.1 that there exists a unique w 2 P�G� (a unique w 2 PA�G�) such thatwjd � wd ; thus w 2 F �G=T � (w 2 F �G=T � \ PA�G�). It follows that j is actually anisomorphism. &

Given the �-group G, set p1 � fp 2 p j G does not split over Tpg, ~p �fp 2 p j expTp � 1 and s2�Tp� � Tpg; from splitting criterions in Abelian groupswe have p1 � ~p and the following propositon holds

COROLLARY 4.2. Given the �-group G, set ~p0 � p n ~p and G � G=T ~p0 . ThenF �G=t�G��WPA�G�.

Proof. Without loss of generality we may assume T ~p0 � 1. By Theorem 2.5 andLemma 2.1, F �d=T �WPA�d�. Now, by Proposition 4.1 (c), w � fwdg 2 PA�G�; theconclusion follows. &

Given a �-group G, for any ; 6� x � ~p0, such that j x j <1, G splits over Tx. How-ever if j x j � 1, it may be the case that G does not split over Tx.

To prove the main theorem of this section, we need a result which actuallygeneralizes Lemma 2.1.

LEMMA 4.3. Given the �-group G of the form G � A� Tx, x � p, for j 2 FA�G=T �set jp � jjGp, where Gp � A� Tp, p 2 x. Then the map g:j 7!�jp�p2x is anisomorphism of FA�G=T � onto C � f�jp�p2x 2

Qp2x FA�Gp=t�Gp�� j jpjA � jqjA

for every p;q2xg.Proof.Given j 2 FA�G=T �, it is clear that jp 2 FA�Gp=t�Gp�� and that jpjA � jqjA

for every p, q 2 x. Now suppose we are given the family �jp� 2 C Then each jp isindex-preserving by (6) and, if we set w � jpjA, we get

Xj=�X \ A�w �Xp

�X \ Gp�jp=�X \ A�w �Mp

��X \ Gp�jp=�X \ A�w�;

i.e. �Xj=�X \ A�w�p � �X \ Gp�jp=�X \ A�w. On the other hand, since Xj \ A ��X \ A�w we have �Xj=�Xj \ A��p � Xj \ Gp=�X \ A�w. Thus Xj \ Gp ��X \ Gp�jp . Then one concludes as in the proof of Lemma 2.1. &


THEOREM D. Let G be a �-group and assume that G splits over T~p0: G � A� T ~p0 .Then

P�G� � �FA�G=T ~p0 � j�Sj�PA�G�:

Proof. By (20), P�G� � �F �G=T � j�Sj�PA�G�, from which it follows that P�G� ��FA�G=T � j�Sj�PA�G� by the remark after Theorem 3.2. By Lemma 4.3 we mayassume ~p0 � fpg. Now, by Lemma 2.2(b), FA�G=T � � FA�G=Tp�F �A=t�A��j , andby Corollary 4.2, F �A=t�A��j WPA�G�. The conclusion follows. &

One may notice that if j ~p0 j <1, in particular if j p j <1, the splitting conditionof Theorem D is satis¢ed.

References

1. Baer, R.: The signi¢cance of the system of subgroups for the structure of the group, Amer.J. Math. 61 (1939), 1^44.

2. Costantini, M., Holmes, C. H. and Zacher, G.: A representation theorem for the group ofautoprojectivities of an Abelian p-group of ¢nite exponent, Ann. Mat. Pura Appl. 175(4)(1998), 119^140.

3. Costantini, M. and Zacher, G.: On the group of autoprojectivities of periodic modulargroups, J. Group Theory 1(4) (1998), 369^394.

4. Dixon, J., du Sautoy, M., Mann, A. and Segal, D.: Analytic Pro-p-groups, CambridgeUniv. Press, Cambridge, 1991.

5. Dugundy, J.: Topology, Allyn and Bacon, Boston, 1966.6. Fuchs, L.: In¢nite Abelian Groups I, Academic Press, New York, 1973.7. Gasparini, E. and Metelli, C.: On projectivities of Abelian groups of torsionfree rank one,

Boll. U.M.I. A3 (1984), 363^371.8. Hall, M.: The Theory of Groups, Macmillian, New York, 1959.9. Holmes, C.: Automorphisms of the lattice of subgroups of Zpm �Zpn , Arch. Math. 51

(1988), 491^495.10. Iwasawa, K.: On the structure of in¢nite M-groups, Japan J. Math. 18 (1943), 709^728.11. Ore, O.: Structures and group theory II, Duke Math. J. 4 (1938), 247^269.12. Ostendorf, U.: Projektivita« tstypen torsionfreier Abelscher Gruppen vom Rang 1, Rend.

Sem. Mat. Univ. Padova 86 (1991), 183^191.13. Sato, K.: Note on lattice-isomorphisms between Abelian groups and non-Abelian groups,

Osaka Math. J. 3 (1951), 215^220.14. Schmidt, R.: Subgroup Lattices of Groups, vol. 14, de Gruyter, Berlin, 1994.15. Zacher, G.: Una caratterizzazione reticolare della ¢nitezza dell'indice di un sottogruppo

in un gruppo, Atti Accad. Naz. Lincei Rend. 69 (1980), 317^323.16. Zacher, G.: Sottogruppi normali ed r-omomor¢smi completi tra gruppi, Ann. Mat. Pura

Appl. 139 (1985), 83^106.


dedicated to h. wielandt on the occasion of his 90th birthday

Documents