Simple groups with prescribed local structure

(1)

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+; <=>=;? @=A<B ?C= >0;B?<D>?E0; 0F BEGH1= I<0DHB JE?C 10>A1 B?<D>K ?D<= 0F H<=B><EL=M ?@H= CAB L==; E;N=B?EIA?=M E; B0G= J0<OB 0F $EP0;/ "NA;B A;M &GE?C 5Q$"&2R/ Q$"&SR7/ JCE1= ?C= H<0L1=G 0F =GL=MME;I A IEN=; I<0DH E;?0 A BEGH1= I<0DH JC0B= 10>A1 B?<D>?D<= EB/ ?0 B0G= =PK ?=;?/ O;0J;/ JAB >0;BEM=<=M E; Q$"&3R% )C=B= HAH=<B H<0NEM= G=?C0MB ?0 >0;B?<D>? BEGH1= I<0DHB JCE>C A<= 10>A11@K5!KL@KFE;E?=7/ JC=; ! EB B0G= <=1=NA;? I<0DHK?C=0<=?E>A1 H<0H=<?@ 5=%I% B01DLE1E?@/ ;E1H0?=;>=/ <=BEMDA1 FE;E?=;=BB =?>%%7 A;M =GL=MME;I ?C=0<=GB F0< B0G= >1ABB=B 0F I<0DHB A<= H<0N=M% +; ?CEB BC0<? ;0?= J= M=B><EL= A; A1?=<;A?EN= >0;K B?<D>?E0; JCE>C @=1MB ?C= BAG= OE;M 0F <=BD1?B/ DBE;I A G0<= ME<=>? AHH<0A>C% .D< AHH<0A>C J0<OB F0< I<0DHB 0F A;@ E;FE;E?= 0<M=</ B0 ?CA? ?C= <=B?<E>?E0;B 0; >A<ME;A1E?E=B ;==M=M E; Q$"&2R A;M Q$"&SR >A; L= M<0HH=M% )CEB ;0?= EB LAB=M DH0; A I=;=<A1ETA?E0; 0F A; EM=A M=N=10H=M E; Q$R%

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U= BA@ ?CA? A B=? " 0F BDLI<0DHB 0F ?C= E;FE;E?= I<0DH ! EB A 1&+)2()1 +)3&14./ 3$3()5 EF E?B G=GL=<B A<= ;0<GA1 BDLI<0DHB 0F FE;E?= E;M=P/ !

#6 $ 6 % " & ' 2 A;M/ F0< =N=<@ 6! 7 % " ?C=<= =PEB?B 8 % " BD>C ?CA? 8 ( 6 ) 7%

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(2)

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6%"

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)C= B=? + ' #;9567 $ 6 % " & EB A ME<=>?=M 10>A1 B@B?=G 0F :5" 7/ A;M ?C= ;=P? 1=GGA M=B><EL=B ?C= B?<D>?D<= 0F =A>C ;9567%

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& *' A% )C= I<0DH B EB >0;?AE;=M E; ?C= B=?JEB= B?ALE1ET=< 0F ?C= 6K0<LE?B #(&6 $ & ' 2! ! ! ! '&% _0;N=<B=1@ >C00B= % % ;9567 B?ALE1ETE;I ?C= >0B=?B 0F 6 A;M/ F0< =A>C &/ M=FE;= ?&% 6 L@ 5(&7% ' (&?&-2% ( I=;=<E> =1=G=;? 0F ! >A; L= J<E??=; D;EaD=1@ AB (&< F0< BDE?AL1= (&% @ A;M < % 6% )CDB

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(3)

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3% &% ? % E 3 QB! F'R (<)' ?E ' E>

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' ?EQB! F'R"?E EB EB0G0<HCE> ?0 A aD0?E=;? 0F QB! F'R B0 ?CA? EB AL=1EA;% fD? E2_{' QB! F}_'_{! F}_'_RF_'_{' QB! F}_'_RF_'_{' E/ C=;>= E EB H=<F=>? A;M} CAB ;0 ;0;K?<ENEA1 AL=1EA; EGAI=B% )CDB ?E_{QB! F}

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(4)

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=<A1E?@ J= GA@ ABBDG= ?CA? !"6 CAB 0<M=< A? 1=AB? \ JC=; 6 L=10;IB ?0 " % &E;>= " EB ME<=>?=M/ :5" 7 ' "

6%"

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&E;>= ! =GL=MB E; :5" 7 NEA ?C= 1=F? <=ID1A< <=H<=B=;?A?E0;/ !2_{=GL=MB E;?0} 95" 7% :=? 0 L= A ;0<GA1 BDLI<0DH 0F 95" 7% )J0 >AB=B >A; 0>>D<%

56 H,+ ).2< 7 % " C (<)+) )?&3(3 6 % " D&(< 6 ( 7 .'1 0 ) QB6! F6R F6 ',( 2,'(.&')1 &' QB6! F6R>

:=? + % 95" 7 L= A;@ =1=G=;?/ A;M >C00B= 7 BD>C ?CA? + % QB7! F7RF7% &=1=>? 6 % " E; BD>C A JA@ ?CA? 6 ( 7 A;M 0 ) QB6! F6RF6EB ;0? >0;?AE;=M E; QB6! F6R% +? EB ?C=; H0BBEL1= ?0 FE;M ? % 0 ) QB6! F6RF63 QB6! F6R% f@ 1=GGA 3%3/ ?QB6! F6R F6 _{' QB}

6! F6RF6 C=;>= + % ?QB6! F6R F6 ( ?95" 7( 0% &E;>= ?CEB C01MB F0< =N=<@ +/ J= CAN= ?CA? 0 ' 95" 7%

96 H,+ ).2< 7 % " C 0 ) QB7! F7RF7( QB7! F7R>

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&E;>= " CAB >A<ME;A1E?@ A? 1=AB? ' ' !$ $/ ?C= I<0DH 95" 7 CAB >A<ME;A1E?@ A? 1=AB? '% f@ H<0H0BE?E0; S%2 0F Q$"&SR/ !2_{EB >0;?AE;=M E; A BEGH1= BDLK} I<0DH 9 0F 95" 7/ 0F >A<ME;A1E?@ '% +? EB >1=A< ?CA? 9 CAB ?C= >1AEG=M

H<0H=<?E=B% !

_.!.::(!g 3%4% 8)( ! #) . =+,4*"(<),+)(&2./ *+,*)+($ 2/,3)1 4'1)+

(.G&'= 1&+)2( *+,142(3 .'1 34#=+,4*3> I% (<)+) )?&3(3 .' &'%&'&() +)3&14" .//$"%&'&() !"=+,4* (<)'C %,+ )-)+$ &'%&'&() 2.+1&'./ 'C (<)+) )?&3(3 . 3&5*/) =+,4* $5'7 ,% 2.+1&'./&($ 'C D&(< . /,2./ 3$3()5 ,% !"#$"%&'&() 34#=+,4*3>

(5)

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FE;E?= BDLI<0DHB% !

(1B0 ?C=0<=G 2 0F Q$"&3R >A; L= I=;=<A1ET=M ?0 I<0DHB 0F A;@ E;FE;E?= >A<ME;A1E?@%

)b".!"'3%]% 8)( 6 #) . -.+&)($ &' D<&2< (<) %+)) =+,4*3 ,% %&'&() +.'G

.+) +)3&14.//$ %&'&()> H,+ )-)+$ 2.+1&'./ ' (<)+) )?&3(3 . 3&5*/) =+,4* 95'7 342< (<.(

2% 95'7 &3 /,2.//$"K6"#$"%&'&()LM

3% 95'7 <.3 2.+1&'./&($ & ' GAP#'! 76&M

S% 95'7 2,'(.&'3 . 2,*$ ,% N5'7C (<) 6"%+)) =+,4* ,% +.'G '>

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0F FE;E?= <A;O7/ A;M ?C=; N 5'7 EB <=BEMDA11@KFE;E?=% )CDB N 5'7 EB <=BEMDA11@K FE;E?= F0< =N=<@ >A<ME;A1 '/ FE;E?= 0< E;FE;E?=% :=? ; L= A >@>1E> I<0DH 0F 0<M=< 3 A;M B=? ! ' N5'7J<;% )C= I<0DH ! EB <=BEMDA11@ FE;E?= A;M CAB >A<ME;A1E?@ & ' GAP#'! 76&% +? EB <=AME1@ B==; ?CA? QB! ;R CAB A; EGAI= EB0G0<HCE> ?0 N 5'7 B0 ?CA? N 5'7 =GL=MB E;?0 QB! ;R ( !2_{% )C=0<=G 3%S >A; ;0J L= E;N0O=M}

?0 I=? ?C= >1AEG% !

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/,2./ *+,*)+(&)3/ i% 9<0DH )C=0<@/ 59 5366d7/ HH% \4]K\]3%

Q$"&SR '% !% $+h.#K '% i% ",(#&K b% &'+)b/ 9,5) 3&5*/) /,2.//$ K3,/4#/)"#$" %&'&()L =+,4*3/ HH% \dKed E; I32<&. !+,4* @<),+$ PQQRC '% fEA;>CE/ -% :0;I0LA<ME/ '% 'A` A;M _% '% &>0HH01A 5=MB%7/ U0<1M &>E=;?EFE> 5366d7% 'A;0B><E??0 H=<N=;D?0 E; <=MATE0;= E1 d IEDI;0 3626%

Simple groups with prescribed local structure

!"#$%& '()*$+ ,"-. /(&+0("1&2 3)04% !-(*0-*(&

!"#$%& '()$+ ,"-. /(&+0("1&2 3)04% !-(0-*(&