Addendum to : a note on primitive groups with small maximal subgroups

ADDENDUM TO : A NOTEON PRIMITIVEGROUPSWITH SMALL MAXIMAL SUBGROUPS Pe e FS s e (1) J .Saxl has p o ided us wi h a p oo o he conjec u e made in he las pa ag aph o ou pape : REMARK . I E is a non-abelian ini e simple g oup which is no an al e - na ing g oup, and i F is a maximal subg oup o E wi h leas n = IE :FI, hen E < H = A n ( he al e na ing g oup on he se E :F o cose s o F in E) is such _ ha he pai E,H sa is ies he hypo hesis o ou P oposi ion . (The case whe e E = An , n '- 5, has been deal wi h in ou pape .) P o o . (Since Saxl's p oo in ol es an applica ion o he O'Nan-Sco Theo em*, we shall p esen a mo e elemen a y app oach .) Co eA (E) = 1 is i media e om simplici y o A n > E . Le 1 ~ K `- A n be no malYsed by E . Aiming a a con adic ion we assume ha E K ; in addi ion K may be choosen as a coun e example o leas o de . As E is simple, E n K = 1, and so G = EK spli s . F om ou choice o K we ge ha K is minimal no mal in G . Since E is al eady p imi i e on he se E :F, so is G . Fi s assume ha K is non-abelian . Le K = S 1 x . . .X S m be he decomposi ion o K in o simple componen e . I m > 1, hen E pe mu es (S1, . . .,Sm1 ansi i ely, and so om ou choice o n we in e ha m ' n . Now IS 1 i n I IS 1 1 m =  IS 1 x . . . X S m 11  IGI I IA n 1  = 2(n! ), which is impossible : he 2-pa o n! does no exceed n-1 2  . (The lame a gumen wo ks wi h any odd p ime as well, so we do no need o in oke he Fei -Thompson- Theo em .) Hence m = 1, in which case K is simple and G `- Au (K) ( o wi hin isomo phism) . Appealing o he Sch eie conjec u e, we ob ain ha E G/K, a subg oup o Ou (K), is soluble, which con adic s he hypo hesis . The e o e K is abelian . In his case p imi i i y o G on E :F oge he wi h minimali y o K as a no mal subg oup . o G yields ha IKI = IE :FI = n . Bu E 1 G ac s non- i ially on K [11  ( ia conjuga ion), a con adic ion agains he choice o n . i *) See E1J, Appendix, o a co ec e sion . 2 5 Thus e e y non-abelian ini e simple g oup occu s as componen o he base g oup o one o he w ea h p oduc s discussed in ou pape . A co esponding conclusion may be deduced om he i s pa o he Theo em s a ed below (a oiding he necessi y o ely on he Sch eie conjec u e) . (2) L .G .Ko ács has obse ed ha he cons uc ion deal wi h in .ou P o- posi ion can be gene alised, and hen yields he mos gene al example : THEOREM . Le H be a ini e g oup wi h subg oups X _and Y such ha X °- Y, X/Y is non-abelian simple, Co e H(X) = 1 _and X `- K whene e K > Y is a sub- g oup o H no malised by X . Pu N = NH (X)n NH (Y) and de ine a g oup G =  E 1 N  H,  whe e E  =  X/Y, o be he wis ed w ea h p oduc wi h espec o he ac ion o N _on E  i en by iewing N/Y a subg oup o Au (E) in he ob ious way . Then G is a p imi i e g oup ; he base g oup E* isnon-abelian, minimal no - mal in G, coincides wi h he socle S(G) _o G, and is complemen ed in G by a co e ee maximal subg oup (namely, by he dis inguished copy o H con ained in G) . Conye íely ; l G in a n imi i e g niin wi h non-ahelian minimal no mal cn lá S(G) complemen ed by a maximal subg oup H, hen G - E % N H, whe e E is he simple componen o S(G) = E 1 X . . .X En and N = NH(E 1 ) has sub- g oups Y = CH (E 1 ) and X = S(N mod Y) wi h he p ope ieslis ed abo e . P o o . The i s pa o his esul can be ob ained by modi ying he p oo o ou P oposi ion app op ia ely . (Obse e ha he hypo hesis ensu es ha Y = CN(X/Y) .) As o he con e se, de ine N,Y,X as in he las pa o he Theo em and no e ha N/Y is necessa ily a maximal subg oup o he p imi i e g oup NE 1 /Y : indeed, i N < No < NE 1 , hen ( o wi hin isomo phism) H < (En N o )1, N H < G = E 1, NH ; c . 161 . Then apply[27,1 .1, in conjunc i . .)n wi h he Sch eie conjec- u e : since Ou (E) is soluble, a g oup as in he las pa o he Theo em can- no ha e a simple socle (see [11,6 .3) . The emainde o he p oo is le o he eade (c . [41) .13 Fo a sligh ly di e en o mula ion o he abo e Theo em (and a di e en p oo , elying on he me hods de eloped in he pape by G oss,Ko ács 133) he eade is e e ed o he o hcoming pape o Ko ács 141 ; see also he pape s o Ko ács,P aege ,Saxl 153, Aschbache ,Sco E1] (whe e he example A5 iA 5A6 N A 5 1 A 6 - he la e indica ing he non- wis ed w ea h p oduc wi h espec o he na u al pe mu a ion ep esen á ion o A 6 - is al eadymen ioned) and o G oss,Ko ács 131 o ela ed and mo e gene al esul s . ACKNOWLEDGMENT . The au ho is indeb ed o L .G .Ko ács and J .Saxl o p o iding he in o ma- ion gi enabo e . REFERENCES . El] M .ASCHBACHER,L .SCOTT : Maximal Subg oups in Fini eG oups . J .Algeb a ( o appea ) . 121 PARSTER : P ojek i e Klassen endliche G uppen . 1 . Schunck- und Ga- schü zklassen . Ma h .Z ., 1984 ( o appea ) . 131 F .GROSS,L .G .KOVÁCS : On No mal Subg oups which a e Di ec P oduc s . J . Algeb a ( o appea ) . 141 L .G .KOVACS : - (in p epa a ion) . 151 L .G .KOVÁCS,C .PRAEGER,J.SAXL : On he educ ion heo em o p imi i e pe - mu a ion g oups (in p epa a ion) . 161 J .LAFUENTE : On Res ic ed Twis ed W ea hP oduc so G oups . A ch .Ma h . ( o appea ) . Rebu el 10 de no embne del 1983 Depa men o Ma hema ics, Monash Uni e si y Clay on, Vic . 3168 AUSTRALIA 27