Simple groups with prescribed local structure

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

.!(8+.-*9:+&+

56 78-()2*0-")86

+; <=>=;? @=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%

96 !"#$%& %)04%%:;<!;1:;="8"-&> ?()*$+6

UC=;!EB A I<0DHK?C=0<=?E>A1 H<0H=<?@/ J= BA@ ?CA? ?C= I<0DH!EB!"

#$"%&'&()EF!CAB A*+,*)+A;M','"(+&-&./;0<GA1!KBDLI<0DH0 0F FE;E?=

E;M=P%

U= BA@ ?CA? A B=?" 0F BDLI<0DHB 0F ?C= E;FE;E?= I<0DH!EB A1&+)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?B8% " BD>C ?CA?

8(6)7%

+;ME<ETT0 M=11V(%W $EHA<?EG=;?0 ME 'A?=GA?E>A XX*% $E;EVV/ *;EN=<BE?AY ME ZE<=;T=/

,EA1= '0<IAI;E [\(/ +K]62S4 ZE<=;T=/ +?A1@%

"KGAE1W HDI1EBE^GA?C%D;EFE%E?

:=? ! L= A;@ E;FE;E?= I<0DH% U= BCA11 >0;BEM=< ! =GL=MM=M E;?0 9'&@G5!7 NEA ?C= <EIC? <=ID1A< <=H<=B=;?A?E0;% :=? " L= A B=? 0F BDLI<0DHB 0F!A;M ABBDG= ?CA?/ IEN=;6!7% "/ ?C=<= =PEB?B8% " BD>C

?CA? 8(6)7% +? EB >1=A< ?CA?/ D;M=< ?CEB ABBDGH?E0;/ ?C= B=?

:5"7' "

6%"

;₉567 EB A BDLI<0DH 0F9% )C= I<0DH:CAB L==; B?DME=M E;

Q$R/ JC=; !'!A;M " ' #6(!$6*'6&% U= >A11 :5"7 ?C= I<0DH 0F

""5,14/.+ *)+54(.(&,'3 0;!%

)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%

!"##$%&'& 8)( ! #) )5#)11)1 &'(, 9'&@G5!7-&. (<) +&=<( +)=4/.+

+)*+)3)'(.(&,'!> ;<,,3) 6 .'$ ',+5./ 34#=+,4* ,% ! ,% %&'&() &'1)? '>

@<)' ;₉567,6J<9_'.'1 ! )5#)13 &'(, ;₉567>

-!..Z% _C00B= A 1=F? ?<A;BN=<BA1@' #(₂!! ! !!(_'& F0<6E;!A;M/ F0<

=N=<@ "%9_' M=FE;= "%9 L@ B=??E;I 5(_&<7"'(_&"< F0< A11 <%6% )C= B=?

9' ' #"$"%9'&EB A BDLI<0DH 0F;₉567 EB0G0<HCE> ?09'% _C00B=?%6A;M (_&%@% )C= GAH#_&!?M=FE;=M L@ 5(_A<7#_&!?'(_A<JC=;A*'&A;M 5(_&<7#_&!?'(_&?</

EB =ABE1@ B==; ?0 L= E;;₉567% )C= FD;>?E0;$_&"6-.;₉567 B=;ME;I??0#_&!?-2

EB A; E;`=>?EN= I<0DH C0G0G0<HCEBG% '0<=0N=<B' /6^$& $&'2!! ! !!'0EB EB0G0<HCE> ?0 ?C= ME<=>? H0J=< 0F'>0HE=B 0F6/ BE;>= Q6^$&!6^$AR'2 JC=;=N=<

&*'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 0F6 A;M/ F0< =A>C&/ M=FE;=?_&%6L@ 5(_&7%'(_&?_&^-2% ( I=;=<E> =1=G=;? 0F!>A; L=

J<E??=; D;EaD=1@ AB(_&<F0< BDE?AL1=(_&%@A;M<%6% )CDB 5(_&<7%'5(_&7<^!%'5(_&7%<^!'5(_&?_&^-27<^!'(_&?_&^-2<

b=;>=%'#^'

&'2

5?_&7^$&/ H<0NE;I ?CA?BEB ?C= B=?JEB= B?ALE1ET=< 0F ?C=6K0<LE?B%

)C= 6K0<LE?B A<= H<=B=<N=M L@ ;₉567c C=;>=/ F0< =N=<@ !%;₉567/

5(_&67! EB B?E11 A >0B=? 0F 6% :=? "%9_' L= BD>C ?CA? 5(_&67!'(_&"6% )C=

H=<GD?A?E0;"^-2!B?ALE1ET=B A11 ?C=6K>0B=?B/ C=;>="^-2!'#F0< A BDE?AL1=

#%B% UC=;>= ;₉567'9_'B A;M ?C= I<0DH 9_'B EB EB0G0<HCE>C ?0 ?C=

H=<GD?A?E0;A1 J<=A?C H<0MD>?6J<9_'JE?C9_'A>?E;I ;A?D<A11@% ZE;A11@ J=

<=>A11 ?CA? ?C= EGAI= 0F ?C= 1=F? <=ID1A< <=H<=B=;?A?E0;&0F!/ EB ?C= >=;K

?<A1ET=< E;90F!^!/ JC=;>=!^&(;₉567% ! )C= F0110JE;I 1=GGA >0;?AE;B B0G= =AB@ FA>?B JCE>C >A; L= F0D;M/ E; A B1EIC?1@ G0<= I=;=<A1 B=??E;I/ E; Q$"&3R% U= IEN= A H<00F F0< ?C= <=AM=<VB

>0;N=;E=;>=%

!"##$%&%& 8)( 6 #) .'$ =+,4*C !'6J<9_'. *)+54(.(&,'./ D+).(<

*+,142( D<)+) '1\.'1 B (<) #.3) =+,4* ,% !> @<)' 2%E'!²'QB!F'RF''56J<F'7²c

3%&% ?%E3QB!F_'R(<)' ?^E 'E>

-!..Z% )C= LAB= I<0DHBEB ?C= ME<=>? H<0MD>?6₂4! ! !46_' 0F'>0K HE=B 0F6% +F J= M=FE;=7_&' #5<₂!! ! !!<_'7$<_A'25A*'&&/ ?C= BDLI<0DHB>A;

L= J<E??=; ABB'#^'

&'2

7_&% _C00B=?!$%6/"'523S7!#'52S37%F_'A;M B=?

4'5?^-2!2!! ! !!27!-'5$^-2!2!! ! !!27% )C=; QQ4!"R!Q-!#RR'5Q?!$R!2!! ! !!27/

H<0NE;I ?CA? 7²₂(QB!F'R(QB!9'R% +; HA<?E>D1A< B²(QB!F'R% &E;>=

!²'B²QB!9_'RF_' J= I=? !²'Q7₂!9_'RF_'% Z0< A;@ IEN=; "%9_'/ ?C=<=

=PEB?B #%F_' BD>C ?CA? 2"'2#% b=;>=/ F0< A11 G%7₂/ 0;= CAB QG!"R

'G^-2G^"'G^-2G^#'QG!#R/ BC0JE;I ?CA? Q7₂!9_'R'Q7₂!F_'R% )C=<=F0<=

E'Q7₂!F_'RF_' 'QB!F_'RF_''56J<F_'7²/ H<0NE;I 527%

+; 0<M=< ?0 H<0N= 537 >C00B= +!3%6/ B=? 4'5+^-2!+!2!! ! !!27/

-'53^-2!2!3!2!! ! !!27 A;M >0;BEM=< ?C= =N=; H=<GD?A?E0;B "'524753]7!

#'52[75S\7% )C= aD0?E=;?E"QB!F_'R EB BEGH1=/ C=;>=E'?^EQB!F_'R A;M J= >A; FE;M.!#%QB!F_'R BD>C ?CA? ".!##%?^E% &E;>= 4!- A<= E;E/ ?C=

>0GGD?A?0< QQ4!".R!Q-!##RR L=10;IB ?0?^E% .; ?C= 0?C=< CA;M ?C= =1=G=;?B

4^".A;M-^##CAN= BDHH0<?#4!]&A;M#[!\&<=BH%/ B0 ?CA? ?C=@ >0GGD?= A;M

>=;?<A1ET= L0?C4A;M-% )CDB QQ4!".R!Q-!##RR'Q4^-2!-^-2R'5Q+!3R!2! ! !!27/

BC0JE;I ?CA?7²₂(?^E% &E;>=7₂² EB ;0<GA1ET=M L@B/ J= CAN=

57₂²7^?E '57₂²7^QB!F'R?E '57₂²7^QB!F'RF''57₂²7^F''B²#

)CDBB²(?^E A;M/ ?0 >0GH1=?= ?C= H<00F/ J= GA@ ABBDG=6?0 L= AL=1EA;%

)C= I<0DHF_'B?ALE1ET=B ?C= B=<E=B QB!F_'!F_'R(QB!F_'R(B% .; ?C= 0?C=<

CA;MF_' EB BEGH1= L=>ADB= '1\/ C=;>= QB!F_'!F_'R'QB!F_'R% )C= I<0DH E"?^E '?^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=;>=EEB H=<F=>? A;M CAB ;0 ;0;K?<ENEA1 AL=1EA; EGAI=B% )CDB?^EQB!F'R'?^E/ H<0NE;I 537% !

U= >A; ;0J =P?=;M )C=0<=G 3 0F Q$"&3R%

)b".!"' 3%S% 8)( ! #) .'$ +)3&14.//$ %&'&() =+,4*> @<)+) )?&3(3 . 3&5*/) =+,4* 9 342< (<.(

2% !²)5#)13 &' 9c

3% ).2< %&'&()/$ =)')+.()1 34#=+,4* ,% 9 &3 &3,5,+*<&2 (, . %&'&() )?"

()'3&,' ,% . 34#=+,4* ,% . %&'&() 1&+)2( *,D)+ ,% !c S% $ $ '9 $ $!>

-!..Z% :=?" L= A;@ ME<=>?=M <=BEMDA1 B@B?=G% UE?C0D? 10BB 0F I=;K

=<A1E?@ J= GA@ ABBDG= ?CA?!"6CAB 0<M=< A? 1=AB? \ JC=;6L=10;IB ?0"%

&E;>="EB ME<=>?=M/:5"7' "

6%"

;_&@G5!7567 EB A I<0DH% '0<=0N=</ L@ 1=GGA 3%2/ =A>C;_&@G5!7567 EB EB0G0<HCE> ?06J<9_' JC=<=''$!"6$% U= H<0N=

?CA? 95"7':5"7² EB BEGH1=% Z0<6% " J<E?=;_&@G5!7567'B₆9₆JC=<=

96 EB ?C= B@GG=?<E> I<0DH 0F M=I<== $!"6$ A;M B6 E;ME>A?=B ?C= LAB=

I<0DH 0F6J<9₆% +? EB DB=FD1 ?0 ;0?E>= ?CA?/ EFF₆EB ?C= A1?=<;A?E;I BDLI<0DH 0F9₆/ J= CAN=

95"7' $

6%"

5;_&@G5!75677²' $

6%"

QB₆!F₆RF₆#

&E;>=!=GL=MB E;:5"7 NEA ?C= 1=F? <=ID1A< <=H<=B=;?A?E0;/!²=GL=MB E;?0 95"7% :=?0L= A ;0<GA1 BDLI<0DH 0F95"7% )J0 >AB=B >A; 0>>D<%

56 H,+ ).2< 7% "C (<)+) )?&3(3 6% "D&(< 6(7 .'1 0)QB₆!F₆RF₆ ',( 2,'(.&')1 &'QB₆!F₆R>

:=?+%95"7 L= A;@ =1=G=;?/ A;M >C00B=7BD>C ?CA?+%QB₇!F₇RF₇%

&=1=>?6% "E; BD>C A JA@ ?CA?6(7A;M0)QB₆!F₆RF₆EB ;0? >0;?AE;=M E; QB₆!F₆R% +? EB ?C=; H0BBEL1= ?0 FE;M?%0)QB₆!F₆RF₆3QB₆!F₆R% f@

1=GGA 3%3/ ?^QB6!F6RF6 'QB₆!F₆RF₆ C=;>= +%?^QB6!F6RF6 (?^95"7(0%

&E;>= ?CEB C01MB F0< =N=<@+/ J= CAN= ?CA?0'95"7%

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

:=? 6 L= E; " A;M >C00B= ?%0)QB₆!F₆R(B₆% +F ? EB ;0? ?C=

EM=;?E?@/ ?C=<= =PEB?B +%! BD>C ?CA? 5+7?*'+% )C= B=? " EB A ME<=>?=M

<=BEMDA1 B@B?=G/ B0 ?CA? J= >A; FE;M 8% " BD>C ?CA? 8(6 A;M +8*'5+7?8% &E;>=B₆F₆(B₈F₈/ ?C= =1=G=;??L=10;IB ?0 QB₈!F₈R B0 ?CA?

? ;0<GA1ET=B ?C= 8K>0B=?B% .; ?C= 0?C=< CA;M/ F0< A;@ IEN=; .%8/

5+.7?'5+.^!7?'5+7.^!?'5+7?.^!'5+7?.% )CDB 5+87?'5+7?8 A;M ?CEB

>0;?<AME>?B ?C= FA>? ?CA? ? ;0<GA1ET=B ?C= 1=F? 8K>0B=?B% b=;>=

0)QB₆!F₆RF₆'2 F0< =N=<@6% "/ BC0JE;I ?CA?0'2%

&E;>="CAB >A<ME;A1E?@ A? 1=AB?''$ $!/ ?C= I<0DH95"7 CAB >A<ME;A1E?@

A? 1=AB?'% f@ H<0H0BE?E0; S%2 0F Q$"&SR/!²EB >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>

-!..Z% &DHH0B= ?C=<= =PEB?B A; E;FE;E?= <=BEMDA11@ FE;E?=!KI<0DH J%

)C=<= EB ;0 10BB 0F I=;=<A1E?@ E; ABBDGE;I ?CA?JEB >0D;?AL1=% 9EN=; A;

E;FE;E?= >A<ME;A1'1=?!L= ?C= ME<=>? H<0MD>? 0F'>0HE=B 0FJ% )C= I<0DH!EB A <=BEMDA11@ FE;E?= !KI<0DH 0F >A<ME;A1E?@ '% (HH1@ ?C=0<=G 3%S ?0 ! ?0

>0;B?<D>? A BEGH1= I<0DH95'7 0F >A<ME;A1E?@'/ JE?C A 10>A1 B@B?=G 0F!KL@K

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% 95'72,'(.&'3 . 2,*$ ,% N5'7C (<)6"%+)) =+,4* ,% +.'G'>

-!..Z% +F'EB A; E;FE;E?= >A<ME;A1/ ?C= I<0DHN5'7 EB <=BEMDA11@K56KF<==

0F FE;E?= <A;O7/ A;M ?C=;N5'7 EB <=BEMDA11@KFE;E?=% )CDBN5'7 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 N5'7 B0 ?CA?N5'7 =GL=MB E;?0 QB!;R(!²% )C=0<=G 3%S >A; ;0J L= E;N0O=M

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

!"Z"!"#_"&

Q$R Z% $":_(&)+::./:,14/.+ *)+54(.(&,'3 ,'!/ !=;M% &=G% 'A?% *;EN%

-AM0NA/55@5366\7/ HH% 24\K2]e%

Q$"&2R '% !% $+h.# K '% i% ",(#& K b% &'+)b/ O5#)11&'= =+,4*3 &' /,2.//$

K3,/4#/)"#$"%&'&()L 3&5*/) =+,4*3/ i% 9<0DH )C=0<@/ A5366[7/ HH% SeSKSd]%

Q$"&3R '% !% $+h.#K '% i% ",(#&K b% &'+)b/9&5*/) =+,4*3 D&(< *+)32+&#)1 /,2./ *+,*)+(&)3/ i% 9<0DH )C=0<@/595366d7/ 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%