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Admissible Direct Decompositions of Direct Sums of Abelian Groups of Rank One

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Admissible Direct Decompositions of Direct Sums of Abelian Groups of Rank One Separatunt PUBLICATION ES MATHEHATICAE TOMUS 10. FASC. DEBRECEN 1963 FUNDAVERUNT: A. O M T. SZELE ET O. VARGA COMMI SSI O REDACTORUM: J. ACZÉL, B. GY1RES, A. KERTÉSZ, A. RAPCSAK, A. RÉNYI, ET O. VARGA SECRETARIUS COMMI SSI ON'S: B. BARNA L. G. Kovics Admissible direct decompositions of direct sums o f abelian groups o f rank one INSTITUTUM M AT H EM AT IC U M U N IVER SITATIS DEBRECENIENSIS HUNGARIA 254 Admissible direct decompositions of direct sums of abelian groups of rank one By L. G. KOVAC S (Manchester) The starting point o f the theory o f ordinary representations is MASCHKE'S Theorem, which states that every representation of a finite group over a field whose characteristic does not divide the order o f the group is completely reducible (see e. g. VAN DER WAERDEN [6], p. 182). A partial generalization of this theorem has recently been given by O. GRÜN in [2], and the main step of the classical proof of the theorem has been generalized by M. F. NEWMAN and the author in [4]. Both of these results arose out of a shift in the point of view: they do not refer to repre- sentations, but to direct decompositions of abelian groups, admissible with respect to a finite group of operators (in the sense of KUROSH [5], § 15). The aim o f this paper is to present an extension of GRON's result, exploiting the start made in [4]. The terminology follows, apart from minor deviations, that o f FUCHS'S book [1]. From [4], only a special case of Theorem 2.2 is needed here: Lemma. Let X be an abelian group, and G a finite group of operators on X: sup- pose that (every element of) X is divisible (in X) by the order of G, and that X has no element (other than O) whose order is a divisor of the order of G. if Y is an admissible subgroup of X which is also a direct summand of X, then Y has an admissible (direct) complement in X. The result of this paper is the following. Theorem. Let A be a direct sum of abelian groups of rank one, and G a finite group of operators on A; suppose that A is divisible by the order of G, and that A has no element (other than 0) whose order is a divisor of the order of G. Then A can be written as a direct sum of admissible, G-indecomposable subgroups, each of which is a direct sum o f finitely many isomorphic groups o f rank one. The proof splits into several steps, and occupies the rest o f the paper. (A) A is a direct sum of countable admissible subgroups each of which is a direct sum o f groups o f rank one. PROOF. Let A = E ( C „ : ) a ) where the C' ordin)al, and 1 runs through all the ordinals which precede cr. Denote the correspond- ing ca, noanircael p rogjecrtionos uA -p- s makes simultaneously the following definitions. Let = s e t (p). I f i is a finite , o f r a n k -oCrdinal and A a countable set of ordinals preceding a, let CI = E ( C l)et Co n e , 2 ba y y , . F/:L) . Eo ) r , a n d i G i s e aa c h n ob r d i n a le pt h s u c he ts h a tm pa o -l l ,e os n et a d m i s s i b l e s u b g r o u p c o n t a i n i n g C . T h e n q t i s c o u n t a b l e ; L. G. Kovdcs: Decompositions of direct sums of abellan groups 2 5 5 as C lisi also countable. Hence C ipthG.e set A Y ' defined by / 1 ' = set().: -< a, C G y , definition provides an increasing chain A Z A . c o u n t a b l e sets i'1 sG y >of- o0rd) inails.s In tucrno, ounen cotnsatrubcts laneot.he r increasing chain by defining its general tgierm Av as Av = U (A T h i s Iet,hL=e f0ol lowfing oproprerti es: i n d u c t i v e 'na v(lAl,) lA °i is- empty. <eb w(Au2) ) ,I f t i s a limit ordinal, Q a , then AQ = ( A frc Uo[poU ( Au; r n t a ab L l(A3y) I f pt-< o e : (tAv1 41 )f Itef A or , • - < m - - a n ofyt UC (Ai O ) = y < ,-V , t h e n Aoe = Crorresdpondiingly, CV =E (C - A, v ; ofndv E A(aCa l l : i( -ft <h eo n Avbu A :(A lAe') C OE s=0 ; A y ) c ov t ) h e :yo - (A2')r i f i s a limit ordinal, a , then CL'= U (CV: v-<o); aw : nd ie if fi f n e s <tf V (A3')p e r — A ; dt < a h n / (.eA4r') ei f 2n, -< a, then C A eavh Q) i n c r e a s 1 ) -Ec e ; ,0e . iM orneovger, e ach CV is admissible; for, Cv is generated by the elements c with a "gs/ C i - i s wc.s E CAic, /ICA'',h an d, aif A E A cg EaAeq tmTo hi n a+hd1m icsvt sibole su ubgnrout p a ' b, land so the Lemma, with X = C'-+ 1 and Y = CA tt'hut at vCoA ,has ianhi a-dmissible complement, say DA, in C ,TA g ei.—v e s 4ah<snand c(fA4o'), ,it stuffice s to prove that A = I (D hj di i r e c A(A3')ewct ha nd ithhe lgeneeral relation Cv v ) which holds for every v with v a . +uA sp a r t :pgTa heA rtva lid -ity< of thisa relation) . is proved by a simple induction: it is valid if v =O. 1biaecaucise of oa(iAl'); ilf it is v alid for the predecessor v — I of v, then Cv = CV oisb ps s u .Ts=n 1 (hI4s ni s v, u i e iw n o f ae-1 mu+ nD _ lDtce1hlimitru omordinr al en sC, A + a gtr bar i ot 1wbt:ayy (iA 2'i). l l ,cu t uhnb e nl p s fersAm sao r l y l o w 0ht a2(oeB ) B—e;ing a direct sum of groups of rank one, A is the direct sum of its maximal f:e-a l r e o m pLCb-sublfgIroups Ap and a torsion free subgroup A ,. The A mn< vf e n p(teh eraeC.oforree av dmcishsibale rsuabgcrout pes, ar nids sot, bi yc th e Lemma with X = A, Y ' A ptAy— runsr t hrough aall primes), A, can also be chosen admissible. Moreover, both A a:p ( r wn h e rd e oaowe1nd thne A p are dkirect sums of groups of rank one. This and (A) make it possible toQ assi ume, without loss of generality, that A is countable and either a torsion free fil) o n )or a tp-gro up. The torsion free case will be discussed first. Gte- e . - (iC) I f A is torsion free and B is a subgroup o f ,hm- d=Decomfispositi on A = A' + A" such that A' is of finite rank and contains B; moreover_ tAe h Ub,oth naAi't aen d rA" a aren admissiblek i subngroup s of A, and are direct sums of groups of rank oeanne. n [i, AsI , t h e n t APROOF. In order to hprove thisa assesrtion, o ne first notes that there is no loss of (s_ uD g),sene ar ality in assum ing that B is admissible and pure in A. The justification of this caan bbe outlined as follows. Let B be any subgroup of finite rank and S a maximal Ec d i r e c t :n s (g va e wD )d t :i; m, vt i: Qh s- .C sE1 ], iC E) b, (l; Dig eE ,df iG :i t, r i es cv ta sl ui md mf ao nr de v i e n r ty hv ep r e c e d i n g a 256 L. G. Kovdcs independent subset of B. Consider the set SG defined by SG -=--set(sg: sE S, g E G); this is finite, f or both S and G are finite. Let 13, be the set of those elements of A which depend on SG; this is an admissible subgroup of A: for, i f a, b EB gG E Ga, tnh edn m a -= m iintegers m, m stso tihat mn[(a — b)g] = .E[(m i g,dependentn m— + m on+n SmG andt hence belongs to BG. I t is easy to see that B is pure in A; moreover, its rank cannot be greater than t he cardinal of SG. Thus kkG) c o sn t a ii n s B a n d B can be replaced by B s,g G Let it be assumed therefore that B is admissible and pure in A. Consider an kni , a. rbitrary decomposition of A int o a direct sum of groups of rank one: gng : i kt(mCl) n A A E ,,=w i.th •the• c,or0r esponding canonical projections y,: A —C A ) , nsAof the inhbdex seot A by A (B, Cl) = s e t : ) -kws;Eu bAase, tn issd B=fin iyted:, eif Sfi >isn a- em0a xi)ma.l ina depend ent subset of B, then B consists precisely n,tslof th uotse behlem sentse oi fa tA w hsich dAe pend on S; so, if O =b EB, then nb = n for some integrs n, n tmt(ei aB s , i l y C l ) (nb)y „= 0 and, as C, is torsion free, n(by„) — O and n = 0 imply that by, =0; so that s=(ts e e n one has A(B, Cl) s e t ( 2 : I E i0a,ti n h a t tAen,e ss Sofy A( B, C„ l).
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