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.
B 3sa tisfy the claims made in (C). U (D) I f A is torsion free, then A can be written as a direct sum of admissible sub- grroups of finite rank such that each of the summands is a direct sum of groups of rank one. + V PROOF. In view of (A), A can be assumed to be countable; moreover, only the case when A is of infinite rank needs investigation. Let A -= E (C ; idirect decomposition of A in which all the C s :ito a(IC r), eA( ca5gn br)e ow riuttenpb as C e' + D' in sucha a way that C, C ' , both C' and D' aore admissible subgroups and direct sums of groups of rank one, and the rank of C'o is fifnite. S upprose that,a fnor sokme p ositive integer n, ot n e . A(hDI ) c c oA r d i= Cnl + Cg 2 + + C " + isa an admissible direct decomposition of A in which all the summands are direct D " stu ms of groups of rank one, all but the last are of finite rank, and C, + A c l + + C. Let 5 be the canonical projection of A onto D", corresponding to ('D1 ). Then C , 5 is a subgroup of finite rank in D", and so (C) provides that D" haas a direct decomposition Dn = C subgroups which are again direct sums of groups of rank one, C nn + + D ' I + " stCi + u c h +d C t,n5 h a t for,A n+1 in place of n. This inductive process defines a subgroup O for each positive O,+ r + inc" te ger i. It is easily seen that the subgroup generated by the O is their direct sum, 1a- n- d iCt c ontains all the C lg aindec ompons1ition sa tdisfynag th en claimds made in (D). +i D.s T(E)h eI f rA eis ftoorsrione fr ee, then the Theorem is true. +v nAo ce PROOF. According to (D), it can be assumed that A is of finite rank. In this '=f a I r cnas e A is trivially a direct sum of G-indecomposable subgroups; it remains to prove te(fhe a0sse:rt io1n ab-ouit th-e structure of its G-indecomposable summands. Let B be an +b arbitraryaoi ) d) G,-imnd ecoimpossable summand o f A, and let B = O. First, a theorem o f Cy BianAER b(Theolnrem e46. 7 in Focus [1]) gives that B is a direct sum of groups of rank onAne. L et B -= + G with all the C di eit+' lem oefn t ofr thae snet okf t ypeos T(Cn ,), ei=1,,... ,n . Put B, —Z(C =t E ( C h iae1= n d aind the elesments of t ype a in B, so that B, is characteristic in B and hence admissible. :lr ,UT ( C e t T:ihTu(s Cthe Lemma, with X = B and Y=B, , gives that B, has an admissible complement ians+ B ; as B is G -indecomposable and B, 0 , this complement can only be O. Hence is )BbnoV B , = so teh at aall the) C, area o f thne sadme type a. )a Bakta In v2iew of ( B ), it is poss=ible to assume for the rest of the proof that A is a count- a=dnble p- grouip. I n (F) a special case will be discussed, and (G) will provide the key tm.oh the gaeneraxl case.i m a ar e lTad )c ;(F) I ft pA =0, then the Theorem is true. htA t ua" PR OOF. If A is finite as well, this statement is trivially true. Let A be countably inhfinite; then A = I (C 1 i -< co) where all the C. are of order p Fo r each positive sd= ineteger j, let Cl = ( C 1 i j ) , and let C AeA 1n Cc' GB b e t h e lo+ s= u b g r o u p +mW gB e n e r a t e d +p bi y Co a+ l l ns B +i 2 Ct . ni T +o h 1n e +l s Di u nk b 'e g ,( r aD o n1 u d) p nh B oa 1 ws c Cb o ,e n +e s Cn i „o s +b tt sa pi rn ee cd i s e l y o f O Decompositions of direct sums of Aai un groups 2 5 9 . the elements of the form cg with e E C, g E G. Then all the Ci and the ClG are finite; C' C ' ' and C i C J G C i s G hold for every j; and U (Cl: 1 - < (o) = A, so that also U ( C 1of A, and since the GIG are admissible, the Lemma (with X = 0 + G1giv:e s that1 each CiG has an admissible complement, say D 0 + 'G. In addition, let D -Gi , < Y = C i G ) holdsI = f orC every j, so that A = U (GIG: - < co) = [ ' (D - < co] = t, o ) 1=E(D E a c h D =fi1n ite,. T h eirenfore eatch hD e Gi i. s T h e n Aisc ubiogsro ru.prsa, eso s thatp theo dniredct idencogmposition of A obtained above can be refined ia d mt i s s i b l e Sdto oi nrei ien wc htich all the summands are finite, admissible, and G-indecomposable. Tiahis renfisnemden t s,atisfi es the Theorem. ns cu m eb ae is ny eo ( G) fLet T be an admissible, G-indecomposable subgroup in the socle S of A. tg o nfiTchenn oT isi finnitte, taend Aal has a direct summand B which is admissible, G-indecomposable, ands whose socle is precisely T; moreover, B is G-indecomposable. oy ei n e d wm PROOa F. It follows from (F) that T must be finite. If T=0, then B =0 will do; ti h a enhence suvyppose that T>-0. Let k be one of the ordinals 0, 1, w ; then pkA is a tn eficharactnreristic ai nd hence admissible subgroup of A. As every subgroup o f T is a dCtirecit suGmm=ahndE o f T, the Lemma can be applied to X = T , Y T A p k A , with yt e t(hee cDonclusion that T npkA has an admissible complement in T. Since T is G-inde- sG -u c1fiomposabnle, it followsi that either TrIpkA= 0 or T npkA = T. If T biIf Tn pgwdA =0e, thenc TrIpkA =0 for some finite ordinals k; but not for all, for -:t 1 e roT mo p o ,-C- p ' " A , il e t m = usi-n thea pforbm inl + I,e and then T w_G _S. incie e_ve,ryj sub) group of S is a direct summand of S, the Lemma can be applied o-_ -to- pX m= SA, ,Y -= T with the conclusion that S T + U for some admissible subgroup fU- . Let U T r I p m " A Ak = — O . imU= nLet it be agreed that w — i c o for every finite ordinal . sp kPut B aiA°ng ,coha in B T=p iB dofA = T . ,so deB of p " i gio. e neIratefd b y Bk and U, „_ k- , krgrHroump 14/IB , is an admissible subgroup in VIB ok f a n _ekis ,a a direct0insd s,ut mhmaend isofr V /Be v e r y a-W/Bd m d dcsk=n iu, r b e gc rt o u p islkoA scIhBosseni is Badmissible, contains Bk, and is contained in p m sto,0c t shue fm : V / b-kofh Blk+e1 c ontiain s T and is contained in S n p m WsBL,e Teu+ Um, , _m a sk.-a u h BukT(1t ' nkwpU , i t h b-Aks1 gs a itkmnX,h fo_llpows =that T' = T + 0 = T. Next, note that p B sAt-ae q.'u ehAnTce; o hf thshe ec h onicesc ofe oB, d e a s T ' km_Vrwe+ 1oo/ B p eS+ i 'UAA) impplie's tnhat b =pa for some a in p " '' I m -kufn+ 1 sp T -mak;= (Bk, U „ , _k_1)1Bk+B k s.a O o n A a ; t f ho re t h i s a , A -_tn,tih isT Bs ehko' wk s _t hati b1 s= pa. = p au +pbn' -= p b' E p B ockdk+ ti k h aes r + B k idkiYhr +m1t m e = d i a t e T-/ B1= E i h V I cB;Wahsm hhaeo/ n nc Ben d B ,- oAek T B =k ksiti a f t fn, + s a o = c -bos E Bs t+)St (h t ao t r e-kBfs s 1pna T, r o u v e,-kti p B o Wip+ ' f db e' s ntkib + n h c e =tkh o h ,tw. n e Ui ,T t h u s Badnel r tk riAh ns n keBo s , fTu _ a E c t fikerc c f ,+b ko ' E B ntdo s o sUk _m+ t eUim r o,, 1e dmnp i tfa ) o. i_al = hon Or nkle 0 ard nd s_m , tes t i u,e k Tvk hn cin . 'ef ea hnt L nro ol a0, e Uyr t k w,s t ,kw hw at V nwh ei yh / _i r t teB B ktc hh hsk k _h ak au„ b lT n - tb/ e Bn d< gB t kp , m rk h nk o, o, e UA na ui n= en pn ,0 kd WV _, ni / kc on B _a wc k ,n sr . 0b t e T ;e ha h hw as e er t - nis ctu etb eg nr o u p B k + 1 260 L. G. Kovdcs
B kall the relevant properties of B o= , If mB isk finite,, this inductive process provides in a finite number of steps a sub- wpgrouip Bt„ h B kBT ;kth ei snod e of B is precisely T. in the first case, every non-zero element of T has height i+kn in A (f1or T - tf h i -i+ p'nB); nhence [e. g. by J) on p. 78 of FUCHS [1]] B is a pure subgroup in A; moreover, s,o -pBp ism bAolu ndead, so thatc a etheore m of KULIKOV (Theorem 24. 5 in FUCHS [1]) implies tc,rh ata B is a direct summand of A. In the second case, every non-zero element of T bo u t f seais oef infinite height in B (as T = p k B Tl c . ,vkn[see e. g. (f) on p. 59 o f FUCHS [ I -]d(pT]ph keoBmre m f18o. 1 rin FUCeHSv [1e]), Br isy a direct summand of A. By construction, B is peadmissible; and the G-indecomposability of T implies that B is also G-indeco mpo- t+ surfi,a blaen. Tnhi ids tco mehpleet es nthec preoo,f o f (G). hA tyka c )o r , d i n g =e (H) I f A is a p-group, then A =E(C,: 2 E A) where each C Bfst oo rthe type) C (p l , Let C= Z(C ita0 h) a t e)=then ,D iiss p reecisietly hthee mra ximcal ydivicsible si ubcgro up of A, so that D is characteristic nBit,n A ahn d is teherefoo re alsor admeissible. Now the Lemma (with X = A, / ,fB: À E A , C i7mtahat oD has anrs adm issible coomplemfent C' in A. Of necessity, C' C , so that C' is o„i , an) direct sum of cyclic groups. Hence it suffices to prove the Theorem under the further .td-o- - D i) pv r of i v isd ei s b da,r ss umcptio yn thcat l A iis ceith) er divisible or a direct sum o f cyclic groups. IelB e A E ia n d fToR (I) I f A is a divisible p-group, the Theorem is true. tD mr= PROO F. In view of (F), the sode S of A can be written as a direct sum of finite, s= E =dapdmissible, G-indecomposable subgroups TA, with 2. running through some index h( eC , cikset A. According to (G), each T is the sode of some admissible, G-indecomposable i,:direct gsu mmand A., of A. Each B onB khA2B, isi s divisible; t o f E h efincen eai cht Be A is a direct sum of finitely many (isomorphic) divisible ,a ir=Agrouaps ofn raknk ,o,ne ( that is, of groups of the type C ( r ) ; by another theorem of BAER, tTl h eorem 19. 1 in FUCHS [1]). The subgroup generated by the BA is their direct nfpC o r hksum, and i t is divisible; moreover, i t contains the whole sode o f A; hence BikC t s e;A =E(B, : 2E A). is(+ or d ) e ni (.1) I f A is a direct sum o f cyclic p-groups, then A is a direct sum of bounded sT)i ; tnad missible subgroups. a)B hkt PROOF. Now all the direct summands of A which are o f rank one are cyclic l„ i eg+hroups. If A is of finite rank, then A itself is bounded, so there is nothing to prove. s pi1In view of (A), it can be assumed that A is countable, so that in the remaining case ofi n i t e rs.A = E(C - < co) where all the C, are cyclic. Let S be the sode of A and S Ti,s otdeh oef C eocach subgroup V of A, the subgroup generated by all the elements of the form vg onai n withca v E V, gEG; VG is always admissible; and, if V is finite, then so is VG. Since s(d, f o r esA is a direct sum of cyclic groups, p''A =0, and so each finite subgroup of A must feu a c a sehave zero intersection with pkA for some positive integer lc ocmh Let k(1h) be the first positive integer for which S s, ruiis characteristicI; in A, it is also admissible. Let St = S 1 pl TpGt +hG( Sn pn kp ( re p:ke ( 11n)" BAot ) ; t h e n S l / 1S =) A = O . voSB , G 1I S 3 i n c e S i,= „( Sn , p k o ) A dUB =i e(k : sB, 1 akB ki s: +i u1 ,v b3 h) g. a. r- sA o- s uk b p-
The steps (B), (E), (H), (I), ( K ) together prove the Theorem. Remark Af t er the preparation o f the paper had been completed, PROFESSOR REINHOLD BAER kindly called the attention of the author to the fact that a combi- nation of results of KULIKOV and KAPLANSKY implies that every direct summand of A is a direct s um of groups of rank one; this would allow some minor cuts in the present proof.
References
[1] L. FUCHS, Abelian groups, Budapest, 1958. [2] a GRÜN, Einige Si.tze über Automorphismen abelscher p-Gruppen, Abb. M ath. Sent. UlliV- Ham5urg 24 (1960), 54- 58. 't3] L. Kov,kcs, On subgroups of the basic subgroup, Pub!. M ath. Debrecen 5 (1958), 261- 264. [4] L. G. Kovikcs and M . F. NEWMAN, Direct complementation i n groups with opeiatos, Arrch• Math. 13 (1962), 427- 433. [5] A. G. KuRos0. Theory of groups, Volume 1., New York, 1955. , [6] 13. L. VAN DER WAERDEN, Algebra 11, Dr itte Auflage, B e r l i n — G ö t t i n g e n — H e i d e l b e r g , 1 9 5 5 . (Received March 15, 1963.)