(OlUliOO J\ ¡\foth c mo t.i co/ J our nol 
\lol. 9, ft.'"!. J, (39 • -4 5). A ¡wil P.007. 

Note s on the Isomorphism and Splitting 
Proble ms for Commutative Modular Group 

Alge bras 

P e t e r D a n c h e v 
13, Ccncrnl l(utuzov St.rcct, bl. 7, Hoor 2 , npn.rL. <1 

t\003 Plovcli v, Bulgaria 
pvdu11ch cv@ynl100.co1n; pvdanc h cv m ail.bg 

A BSTRACT 
h is provee! Lhat. t.hc p-Splitti ug Croup On.sis P roblcm fo r a rbi trury ubclin n 

grou ps nnd thc S pli tting Croup Busis Problcm for p-mixed groups Rrc, in foc t., 
c c¡ui\'n lC'.nt in Lhe com1m1 1.ativc g rou p nlgcbrn o f chnrnclcri:slic p. In nddi t.ion , n 
ncw schcmc o f o btni11ing t hc complete set of invnrinnts for such n g ro up nlgcb ru 
of n p-splíHing grou p is givou. In pnrticu lnr, ns npplicMion , thc full systom of in-
vnrinnt.s fo r a gro u1> nlgobru whcn thc group is p-s plitling Warficld is cstnblit1hcd 
ns wc ll. This s upplics own rcccnt resulls in (Bol. Soc. Mal. t\l cxic111111 , 20011) 
nnd ( Acln t\lalh. Si nicn, 2005). 

RESUME 
SC' pruebo. que el Problonm de Doses ngru1>ndns p-dh•idid n p arn grnpos 

nbclianm nrbi t rurlos y ol Prnblcmn de BRSCS ngrnpndns divididn ¡mm grnpos 
rnl'Zl:lndm 11 es, c u CÍt..."Clo, cqui v11lcntc e n el filgcbrn de gTupo coun111t11tiv11 d e 
cnrsclerfst icn 11. Adcm ÁN, HO e nt rega un l'tiqucmn nuevo parn obt.encr el conj unto 
completo de invnrin nt cs parn e:m filgcbrn d e grupo de un grupo 1>-dividido. Eu 
porticulnr. romo nplknción, se c:;tnhlL>co ndcmfi.s el s islemn comploto de invnri-
nnt~ paro un álgcbrn ele grupo cmmdo e l gru¡>0 es \\'l\Tfield ,,..div idido. 8sto 
e nt ttga rcsult.ndos p roplo11 y novcd O!KlS e n (Bol. Soc. Mal. Mcxicntm, 200•1) y 
( Acln Mnth. Sinicn , 2005). 



•10 Pctcr Danchcv 

K cy words and phrosos: i.~omorplu.m1 problem. p-6Jlf1fhng group.f, p -mued 
ymu71/j, ll'arfidd group.~ • .f•mply prc_,c11tc:d grou¡11. 

Muth. Subj. C loss.: P rinwry ROC01, 1653,, eoK eI : Secondo ry J6U60. 

1 Introd u ction 

Throughout lhis brief puper wc shall m;c, in a traditional mnnnc r, thc lcttcr FG ns 
thc b'TOup algcbra o f an abclian g:rou p C writtcn multiplicati vcly ovcr a ficlcl F. Por 
such a group C , 1hc symbols G,. a nd G1, urc rcscrvcd for its ma ximal to rs ion subgroup 
(= torsio n pnrt in othcr tcrms) a nd 1>-primary cornponent, rcspcctively. 

The Splitti11g Problem for arbitrnry mixcd g roups1 going fro m i\lny [10] ru; a spccial 
pa rt of the / somor¡1hism Problem, as ks of whethcr the separntion of G, as a dircct 
factor of G can invarifmtly be rotricvcd from the group algcbra FG. Unfortmrnt.cly, 
this qucstion has a ncgative scttling in general firstly provcn aga.in by li.·lay {scc, for 
cxa mple, ¡t2J). r..lajor advantngc in various aspects on t hc prescntcd t.hcmc was done 
in ll J {sec [5] t oo for commu1;ntivc grou p ulgcbras ovcr ali fielcls). 

As was firs t.ly obscrvcd and conjccturcd in [13], t.hc wca k \•crs ion of t.hc Split· 
ting Pro blcm fo r 1>-mixed g roups can , cvcnt ually, t o have a posit ivo solut.ion ovcr thc 
commulativc group algcbru with prime charnct eristic 7J. f'or this purposc, li.foy, how-
cvcr, assumcd t hat thc Direct Factor Problem fo r p-mixcd g rou ps is true. In whut 
follows, wc s hall s how thut. thc :;nmc cnn nlso be cxpccted undcr t.hc vnlidity of thc 
lsomorphis m Problem for 7>-mixcd groups. 

Thc difficu ltics cncountercd in thc thcory o f mixcd s plitting abelia n g roups can 
bccomc decidcd ly lcss complcx, if it is po&;iblc to reduce thc qu ·t.ion t.o s plit.t.ing 
mixcd abclia n groups whosc nu.1.ximal t.ors ion s ubg roup is ¡>-primary, t.hat. is e, = e,,. 
Call such an a b cli a n grou p a v-mi:i:e<l gro11p. In t.hc next paragrnph wc s how t ha 1 
lhc p-spliu ing problcm fo r fl mixcd group is reducible to t hc samc problcm far p-
mixcd groups. Aft cr this, wc [ook at ccr tflin cillSScS o f f>-Splitting groups for which 
lhc com m ut ati \•c modula r g roup algcbrn with chnract eris tic p posscsscs a complclc 
set. of irwarinnt.s. 

2 !Jai n Theore ms 

As u.sunl. thc mixed abclion group i:s sflid Lo be ¡1-svlitting, rcspecti vcly .SJ1litting, if 
GP, rcspcctivcly G,, is n dircct fact or of C . 

Bcfor p rcscnt ing t hc ce nt.mi rc:sults, wc n cccl lhc following tcchnicnl nsscr tion 
which is thc crucial point. 

Lc mnm. LA!t C be <m abelian gmup . Thcn G Is p-spl1ttmg <====:> G / U G q is 
spltttmg. 

.... 
Pro of. First of ali, not e tlmt (G / 11 Cq)1 = G,/ U Gq = (C,. x 11 Gq)/ U C q ~ 

1¡i¡t11 '11-P qr¡i.p qfl p 



Not.cs o n t.hl! lsomorphhm l nud Split.Ling P roblcms for . •11 

e:,,. 
l. º=;-" : Writc C = C 1, x M fo r somc M :5 G. Thcrcforc. it is self-cvident t hn L 
G/ ll e,,= (G, / ll c,,)(M( ll e:., )/ ll G,,) = (G/ ll G,).(M(ll G,,)/ ll G,,). 

,,,.¡.,, ,,,,,_,, <¡ji:p •1 rf.:.1• ,,.,¡,,, '*"''' 'l~l' 
Wlml. rcnm ins Lo dcmons LrnLc is LllllL Lhc int.crsccLiou bet.wccn thc t,wo fnctors is 
equul Lo OHC. This, ccrt11 iu ly, is uccomplis hcd by s howing thnl e, n (/1/ lle,,) = 

Il C:,1. l ndccd , wit.h t.hc nid of t.hc modular law, wo cnlculntc thn1. 
1¡rf.¡1 

e, n (M il e,,) = <ll G,,)(G, n M) = <ll G,,)((G,, X M, ) n JI/ ) 
•1rf.:.1• q rf.:.1• q rf.:. ¡> 

= <ll G,,)((G,, X ll M,,) n M) = <ll G,,)(G, n ¡\/ ) 
•1# 1• ,,,¡.,, ,,,,,_,, 

= u C,,, so com p lcting Lhis parL-lmlf. 
,,,¡,,, 

q rf.:.¡J 

2. "*=": Writ.c e/ U e,, = (G/ u e,,), X (N/ U e,,) or cquivnlcnLly G/ll e,,= 
•¡:h• q rf.1> 'li-l' •1 rf.:.1• 

(G,¡ LJ e,,) X (N/ ll e,,) fa r sorne N :::; c. Conscquent.ly. s incc N 2 u e,,. it, 
•rli1• ,,.,,,, 'l'h• 

is noL lmrcl to clct.cCL t1hat. e = GtN = c,,N wi Lh G1 n N = u'l,,'I' e,,. Menee 
G¡, n N s; cu G,,),, = l. T ltcrcby, e: = e,, X N nnd WC ore finis hcd. 1 

,,,,,_,, 
.J. Oppclt. has cons iclcrcd in (j 11!]1 p . 1260, Corollnry) anothcr imcrcsLing rcdl1cLio11 

of Lhc split.Ling mixcd g ronps to l.he p-sp li tting oncs. 
T hc fo llowing cquivn lcnce p lciys a key role in our furt.hcr ex plorn t.ion . 

P ropositio n . Let G be. 11.11. nbe.lúm grou11 (lll(f F a field o/ char( F' ) = p. The.n the. 
following t wo implications m 't'!. 1iq1L'iu<1Üml: 

( I} 1/ G' is p-mixe.tf svN.l.f:ing anrl FG 3'! FN <IS F -algcbros for som<~ fJl'Ott71 H => 
¡.¡ fa v-mi.xed splitti119. 

(!!) 1/ G is p-s11litt.in9 nnd FG 8=' F N t1s P -algcbms for somc grn11p H => N fa 
1'-·~fJl'ilf.ú1g. 
P r oof. Thc lcft. implic nLion ( 1) => (2) follows likc t.his. Choosc e t.o b e p-spli LLing. 
Sincc, in vicw of j !GJ, tho F-isomo rphism bctwccn FG nnd F !·!forces nn F-isornorphis m 

betwce11 F(G/ lJ Cq} ond F(N Jll N,1) 1 wc nppcol Lo t.h Lcmmn to concludc Llmt N 
'li-P <¡rf.:. ¡1 

is also />-Split.ling. os stat cd. 
T hc right implicnt.io n ( 1) <= (2) is s t.rnig htfor wnrd bccnusc lhe spl iLt.i ng grou ps 

m e known to be p-split.t.ing 11 11 d ulso t.ho p-mixcd 1r-s pl itting groups urc splitt.ing. 1 
As alrcncly inclicnt cd 1tbovc, W. 1\1lt1y s howcd in [13J lhnt if Lhe DirccL FacLor 

Problc1 n for p-mixcd g r011ps l1old:. in tl1c 1111\nnativc, t.hcn thc samc is true fo r ~l tc 
Split.Ling Problcm for ¡r111ixctl groups; 11s wi ll be clemonslrnlcd in t.he sequel ( for 
uxtu np lc our Mnin Thcorcms lnvurinms Sllll cd bc\ow) t.hc posilive solu tion of t he 
l:;omorphi~m Problcm íor p- rn ixod grou ps as.sures nnich more, nnmcly t.hu.L both Lhe 



42 Pe! cr Donchcv 

¡rSp lil.ling P roblcm Hnd Lhc ]::;0 111 o rpbi:m1 Problc111 ío r nrbit rary ¡rspli t.t.i ng groups 
hold posilively. Bes id cs, wc clHim Llmt. t. hc well-known Dircct. F'nct.or Problcm far 
1>-groups. which is wc11 kcr tliun Lhc DirccL Factor Problcm fo r ¡r mi xcd gro ups, is 
nbsolutcly enough fa r t.hc fu!l rnsoluLion of t he ge ne ra l p-Split.ting Problem (n sketc h 
o f prooí was firs t g ivcn in [21 but t hc co mplete confirnmt.ion is n problcm o f unoLher 
rcsearch art icle whcrc n ncw 11pprooch 1ni ght. work). 

F'orcmost., befare bcg i11ni11g witb Lh e nwin th corem , for t hc co nvc ni cnce of t. hc 
render, wc rccollcc t. somo more qucst ions in d etn.ils. 

lso m or phi s m P roble m for v-M ixed G r o ups. Let. e be a v- mix cd Hb cli nn group 
nml lct F be a ll e td o f clmrncterist ic p. Docs it. fo lt ow that t.he F-isomorp hism betwccn 
FN nnd FG for any group ¡.¡ implics t hat ¡.¡ and G are isomorphi c? 

I som o rphi s m P r o bl e m for p- S pli tting G r o ups. Let. C be a p-sp li tti ng ub cli nn 
group nncl lct. F be a fi cld of charucterist ic p. F'ind t hc co mplot.e syst.em oí invnri1111 ts 
for FG. 

In t he next. lines, we sha!I i!lustrntc t.hat, the positivo answe ring o f !.he Isomorphi sm 
Problem far p-mixcd gro ups yie lds Lbc pos itivo solu tion oí Lhe lsomo rphi sm Problcm 
far p-.split.Ling groups . Ancl so, wc ll ave acc umu\ated ali Lhc infonna!,ion neccssnry to 
procccd by prov ing t hc following. 

Ma in Theorem (lnva l'ia nts). Su p¡wse G i.s an abelian ¡¡- s plitting g1Ym7J and F is 
a fi eld o/ cha1·( F) = p ,¡:. O. Und e1· /,he assumption o/ ln1thful11ess o/ lit e /so m.orphüm 
Pmblem for p-mi.xed gro1Lps, F H 9:;' FG are F -isomor7Jhic for so m e gr011p ¡.¡ if rrnd 
011/y if ll1e following conrütions hold: 

( 1) H is p-svlit.ling nbelian; 

(2) H,,"' G,,; 

(3) H / H, 9< G / G,; 

(4) F( H / H,,)"' F (G/G1,). 

Proo f " n eccss ity ". Accordin g to [!6] we o btuin t.hat F (G/ IJ Gq) ~ F ( N/ U Hq) , 
q"#-p q,¡. ,, 

whc ncc \·in our ass umpt. ion G/ Il G,1 ~ H JIJ N,,. Thus CP ~ (G/ il Gq), ~ 
</ # 1' q'/ p qr¡,. ,, 

(1-1/ U flq)t ~ l-J1,, i. e. (2) sustnincd. J\forcover , refcrring to thc Lc mm a, wc infcr 

''" tlrnt ( I) is true. F'url.hcrmore, (3) fo!lows fr om [!O] (scc [<lj too) as wcll ns (•I). 
" s u ffi cic ncy 11 • V·/ritc e 9! e,, X G/ G,, oncl /-/ ~ flp X H / H,,, hcnce FC ~ 

FCp ® F F(G/G,.) 1111d F H 9! FH1, ®i•· F(/-1 / H,,). Conscqucmly, (2) a nd (.I} cnsurc 
thnt FC~ F/-1 , us nssc rtcd. 1 

Wh cn F is nn 11lgc brai c1dly closcd fi cld , in nccordnnce with IJ Jj, Lhc iso rn o rphi sm 
(11) cnn be replnccd by (111) IH¡/ /-/1,1 =1Gt/G1,j. W hcn F is an nrbit rnry llclcl but. 
lhC quoticnl G,/G1, is finito, owin g fo r instnncc to j2J, lite isornorphi sm (ti ) rn ny b 
rcplnced by (4") 111 ?' / 111,I = IG~ 1 / G1,I Y q ,¡:. p nnd i E s.,(F) U (O} whc rc s 11 (F) = 
{i E N: F(11,) # F(111 +i)} miel 11; is t.hc prim ilive q'-th root o í unity tllkcn in thc 
nlg brnic c losurc o f F. 



,ll'lll:l!!l 1 otM! on l hl• h;omorphis m nnrl Split.ling Problcm:, for ... 

3 App licat io ns 

l lcn' wc shnll now considcr concrct.c clnssc:; of grou ps íor \\•hkh 1 he fo. [nin Thcorc111 
ln v1.1rinnls holds. llowc\'Cr , wc bound our at.lcnt.ion onlr on \\"arficld g ro11ps 11\tho ug h 
t.hc sm nc rcsults ore vnlid cvc n fo r t hc m ore lllrgc clnss of ¡1-clcmcn1 ory 11' -grou ps, for 
1111 ordlnnl numb r ¡1, clcfi nccl 1111 d st.ud iccl in 1 J. \Ve cmphMizc tluH gcncrnl mct.hocls 
o f ol>tai ning thc i n vnrinnls o f s pl it.Li ng com m utntivc group nlgcbrns wcrc fir~t givc11 
in [2[. 

T hcor c m 1 ( l n va r ia n t.s). Suppo.~e llwt G is a f)·spl1tt111g Warfirld f1bclio11 group 
Oltd lhat !( i.s an algebmicolly cloHed fidtl of cJw,.(I< ) = 1' >O. Tlic r1 /( 11 ~ f( C ru 
/( -olgcbms /01· rmothc r ,qroup 11 ·if tm.<l 011/y if ll1c /ollowmg n:md1tfons fl1Y' f11lfillctl: 

( 1) 11 is p-.~11lt lli11g nbcfia.n¡ 

(2) 11."' G,: 
(9) 11/ 11, "'G/G,; 
14) [11./111,I = ¡G,/G,,¡. 

Proo r. In IGJ (cf. [7] 11s wc ll ) W C llave provcd thnt c¡ LJ cq ~ 11111 11,,. Tl111l. h; 
qj.p q>f-p 

why, Lhc Mnin Thcorcm npplics Lo d e ri ve thc cle;ircd clnim. 

T h cor c m 2 ( l nvu ri n n ts) . S npposc lltat e 1$ a s11litt111g Warfield abelian fJIVUp wiU1 
11 jitiile factor G,/C1, cmd thal. F is rw llrb1lrnry firld o/ chor( F) = p > O. Th c11 
/·' 11 ~ FC as F -11l9cbm.~ for nnolher gro11¡1 11 1/ and only 1/ lh~ foffowi ng c011ditio118 
o,.c 1"Cal1zcd: 

{ I} 11 1s ,:,;p/1ttrng nbrlian; 

(2) 11, 9! G,; 
(9) 11/ H, 9! G/ G,; 

(4) IHf / //,[ = !Gf / G,,I, Jm· all ¡16mes q #- /> aad i E sq(F) U (O). 
P r oor. Bccnu.sc in vir Luo o f [G] (cf. 17] too) W(' hn\'(' dcduccd thnt c ¡ LJ G'1, ~ 

q t/;¡> 

¡.¡ f li llq, wc cmploy ni o m.:c Lhc rvlniu Throrcm to gct thc '''fmtcd c lnim . 
. ,-;.,, 
Wc l('rminnt th(' pnpcr by t.hc hclpíul obM-rvnlions thel although (,' und G'/G'1, 

urc both Wo..rlicld. // nnd ll / 111, urc noL 11cccssnrily from lhi.s grou p cl n.ss. Also, s iucc 
(;1, bcing an -group (:.ce o.g. [O] for the ,rlocal cose or ll5J for thc g lobn l 0110) ncccl 
uol be simply pre;cn1ccl, Lho ln!"!L two thror ms nrc nL firsl look indcpcndonL fro m [3J. 
llom·vcr. bocatL..C Cp f\8 bci11g H dirc.•ct fo(_·t or oí Lhc Wnrlicld grou p e is nl:so \oVnrfic ld, 
ho ucc simply p~nlccl , Theorc ms 1 nnd 2 nrc d educible from l3J. 
R o n mrk In fá, ¡>. 160. lin(' 8(-)] thcrc is n misprint, 1mmt'ly thc farnu d n T(V P [G'j) = 
V P[T(C')l ,hould be writtrn "" T (I ' P[C'J) = 7'(\I P[T(C')i):""" also [•I/. Morcovcr, in 
[2, p . 11. line l l{+)J t lw " ifr' 11111st b r<'nd ns "if'. 



11 Pctc r Dnn chcv 

Recei ved: Oc t 2005. Revisad: Oc t 2005 . 

R e fe r e n ces 

[t] Z. C llATZ IDAl\I S 1\N D P. P 1\PP1-\ S, On tli e spli tlt ng gro 11p bosi.s pro blt> rii for 
abelra n 9m1qJ rings, .l. Purc Ap p\. Algc bra (1 ) 78 ( 1992) , 15- 26 . 

[2 J P . ÜAN Cll EV, lsonw17Jhism o/ commutat.ive group algebms of m ixe<I splil ting 
groups, Compt. rcnd. Acad. bulg. Sci. ( 1-2) 5 1 ( 1998), 13 16 . 

[3] P . DAN Cl rn V, lmmrinnls for gmup algebrns of abelian gf'O ll 7JS wilh .~i mply 
presen ted compon enl.s, Compt . rcnd . Acad. b ulg. Sci. (2) 55 (2002), 5 8. 

!ti] P. DANClll~V, A n ew simple ¡nvof o/ lhe W . Moy 's cloi m : PC <Lel.cn n i" es 
C / G0 , Ri v. tvla t . Univ. Pa rm a 1 (2002), 69- 71. 

IS] P . ÜANC HEV, ls omo17Jhis m of comm.ul ative g,.0 11p alg ebrns over all fiel ds , 
Rcnd. lst it. J'vl a t. Univ. 'l''ri cstc ( i-2) 35 (2003), 1•17- 1611. 

[6J P. DANC ll EV, ;t n ote. on th e isom.01·phis m o/ m ociu lm· gmt1p olgebni s o/ 
global W ar f iel rl abelian gro1tps, Bol. Soc. Mn t.. lcxicann ( 1) 10 (2 0011) , <19 
5 1. 

[71 P. D ,\N CJrn v , lso m orphic comm11 tat.ive gmup algebras o/ p- m ixcd \\lmficld 
9rot.1 J1 S, Act.a !vl a.tb. Sinica (4 ) 21 (2005 ), 9 13- 9 16. 

[Sj P . Q ,\ NCll EV, On /Ji.e iso m. m·phic group alg cbros of ;sot.ype .rnbgro11p .~ o/ 
lllor/ie fd rJ.belúm. grou7JS, Ukrait 1. ivlat. h. Bull. (3) 3 (2006) , 305-3 14. 

!9J R . H UNTEll AND E . WALKtm , S-gro ups revisit e<I, P roc. A mc r . 1\l at. h. Soc . 
( 1) 82 ( 198 1), 13- 18. 

l!O] \V . r>. IAY , Com.11ut l<i li-ue gm u p algebras, 1Tuns. t\ mc r . r>. lat. h. Soc. ( 1) 136 
( 1969), 139 1,19. 

ji lj \V . tl. IAY , ln v m·úm.ls f 01· co 11 mwtative gro up algebrn.'i, llli nois .J. Mnt.h . (3 ) 
1 5 (1 97 1), 52:> 531. 

[12J \V . Lw , /so m.0 1·ph:ü m. of gmup olgt>brns, .J. Algcb ra ( I} <10 ( 1976), 10 18. 

!I JJ \\" . ~IAY , T he di.rnct f actor probfem f or modular abcli o n gro 11p algebros, 
Contemp. t-. l nLh. 93 { 19 9), 303- 30 . 

p .1¡ J . ÜPPE l.T, Ali.xetl nbelilmgro 1111s, Can .. J. Mnth . (6) 19 (1967), 1259 1262. 

l15J R . STANTON, Wmfre/rl grouJJ s tmd -gro111M, (prc print.). 



,«ffl!!!u ot~ 0 11 t lw b o111nrpl1is111 nncl Spli Lting P roblenL-. for . 

[IG] \\'. LLE!t'i' , A ron;crtwY' rcfot1119 lo /11(' 1somorph1sm J1rY1bfcm fo1· comm:u-
tatwc grou¡' algebm.t, in Croup nnd Scmigro up Rinp.. North- 110111111<\ t-. lnth. 
Studi~, No. 126, o rtl1- l loll1111d, An1SlC'rdnm , 19SG, 2·1i 252. 


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