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From the Preface to the First Edition
ix
FROM THE PREFACE TO THE FtRST EMTiON a "mhg ahe g e n w z e d hingd. Roughly dpoken, a nem-hing ( N , + , . I , &ehe + i.6 not nece.6bafdi.g a b d i a n and wikh o n t y one dinMblLtive
Nm-hhgd
bw,,.
Neah-hingd ahide i n a natuhae way: &tabe t h e be2 Mlr) 0 6 att M p p h ~ g d 06 a gmup (r,+l i n t o &e..t.ed, de6ine additian + point-whely and o a6 cornpabLthn. Then (M(r1 ,+, 0 ) i.6 a n m - h i n g . Even i 6 r h a b e l h n , o n l y one disthibutive &UI iA aeUayb 6utd.ieeed: ( i + g ) q h= 6oh+goh holds by t h e de6inhXon 06 6+g white doh 6o(g+h)= dog+deh we mutd have to adbwne t h a t 4 i.6 a homomohphibm. Anothm cxrunpte i.6 d U p p ! i d by t h e polynonciaed w . 8 . t . addLtion and hub6.titLLtion. A WeU-known tlebUet i n hing theotly dayb that e v a y hing can be embedded into t h e hing Elr) 06 a l l endomotlpL.bm 06 borne a b U n gmup r. Foh n m h g b we pmve ( 1.161 that e v a y nem-&g can be embedded into M(r) 60% dame gmup r. Hmce one migkt v i w hing theohy ab t h e " f i n m t h w h y 04 gmup mappingb", while n m - h i n g b pmvide t h e "non-.&npAIL thwty". SUILpILi6ingly, a t o t 06 ".&nm tle6uLtb" can be L t a n b 6 r n e d ~ t ot h e gmnehae c a e aiteh dLLitabte changes. Foh i n b m c e , t h e "aXomb1' 06 hing theohy, t h e phinritive h i n g d , m e d a d b e d by t h e ~auwusd e n b d y theohem 06 N . 3acobbon (doh h g s w i t h minimUn condition: W e d d m b m - M h t h w h m on bimpte h i n g d ) . foh nean-hiftgb bAni.kb tlebbcLetb cancaning phimit.iue neat-hingd wme 0 M a i n e d v i a t h e u m k 06 6QvQAd aLLthohd ( b u t t h e pmo$b aht to&d..ly d L 6 6 a ~ n t )t:h e hole a$ ffo%(V,Y) doh aingb iA played by M(r) oh dome hekzted t y p e s i n the n m - h h g m e (4.52, 4.54).
t d i s t O h i d y , t h e 6 h t 6tep tolllaha neah-hingb wab an a c O d C h e d Q W h done by V i c h o n i n 1905. ffe dhowed tfmt thehe do ex#t ' ' ~ i c t d bw i t h onty one d i s ~ % M v e W ' ( = n m - 6 i d d 6 ) . Some ylatch thebe nearr-dieeds & h o d up again and phDued t o be uae6LLe i n c o o k d i d z i n g cehtain impohtant daAbeA 06 g w m a c p . h e d ( h e m that lk6cahte'b method 06 coohdinatizing t h e " u b d " pkkne by t h e d i d d 06 heal numbm wa6 one 0 6 t h e m o d t ducceb6~u.t d t e p b .in geomeZq). It wa6 Zabbenhaus who wa6 abte t o d & W e all ~i.n..Lte n m - 6 i e C d ~ ( 1 . 3 4 ) . ~ U m d a y b ,n m - 4 i e l d b ahe a nrigkty tool h~chahactdz&g doubly thanbLtive gtloupd ( 8 . 4 4 1 , inincidence ghoups (8.6%) and Fmb&us gmupb ( 8 . 8 1 ) . Since t h e dwn 06 A3uo endomohppkibm 06 a n o n - a b m gmup ( r , + )i.6 not an endomotlpki6m i n genehat, t h e d e t d E ( r ) 06 aU 6 W e 4wn6 and di6?4menceb 06 endomohphh6 0 6 r m e consklehed. hebpect t o addition
Preface
X
and compobLtion, t h u e E ( r ) '6 ahe neah-hingb belonging t o t h e C h b 06 t h e "dib.thibU.t.iw&y gwnehated" neah-hingb Many p#u2 0 6 t h e we,U-ehtnblhhed fheohy 06 hingb wehe tRand5eNred t o n m hGzgd and n w neat-hing-bpecidic ~ e a t w uw4he dincovetled, bLLieding up a theoay 0 6 nu&-hingb b t e p b y b t e p . Up t o now, about 550 p a p m on Wah-hingb (and n u - d i d d s ) w i t h about 8000 pageb appeahed i n phint, b u t t h e m exha2 no book on t k i s b u b j e o t . Tkid book .thieb t o unidy t h e theohy and L t h feminology and t o g i v e a bgbtematic and w&-asbohted account 06 t h e phuent n t a t e 06 t h e t h e o q . Some aemahn am .to be made: ( a ) GenmaUy, 7 avoided t o g i v e ph006b doh t h e o h m which ahe &hm not &ng t h e main b . & m 06 dibcudb-ion oh a h long ~ ones which contain bpeciae mcthoh seemingly a,oflicable o d y i n .thLh context, cannot be b h p t 4 i e d b y pewhiow hunuets and invo.P,ve many o t h m ( e . g . geornethicat) dC?.tdied, b u t ahe heaiLiey accubib& in t h e f.Ltaatwre. ( b l Sevmak? hulLetd 6oUowing dhom uvLiwm& d g e b m oh &om t h e t h e o h y 0 4 ghoup with mubkple opnehatou ahe c i t e d , bu.t not p v e d i n a d m to dev,i.de hes1Letb which ahe bp&fiic doh n m - h i n g b and Rhobe which ahe
.
not. @L m y 6~wmbeing a m a e coUecLion 0 5 - t & w i d hesuets concaning dome "pathologiCae" b y b t m tU.ithout any application t o o t h m bzmches 06 mathematicn. A p a h t dotun t h e appfic&onb concaning axiomaticb and geomethy mentioned above, bpeciae & b b u 06 d i n i t e n u - h i n g b ( t h e AinLte "plans" nea-hingb) g i v e nw and highey eddi&nt C h b u 06 balanced p a h a w e A m ( 8 . 1 1 7 - 8 . 1 2 4 ) . Moheovm, incomplete block debigm aeheady [uith b& these p l m neah-hingb can be uaed t o chaaactmize Fhobg m u p , hence a h o 6.inite ghouph w i t h d i w e d - p o i n t - d u e a&tomohpkism ghoupb ( 8 . 9 6 , 8.971. 16 r i d a &inite, inwaAhaXy a h p l e n0n-abcL.h gmup, E ( r ) h "phimctive" and t h ee 6 o h e ewmy belb-map 06 r &ixing z m h .thk " b u m 0 6 endomohpkism" [exact @munLLeation i n 7 . 4 7 ) . Anotha v m i o n 06 t h e d e n b a y theohem 4.52 b h o u t h a t t h e d m a y p m p e h t y h ( i n t h e n e a - h i n g - c a e ) bometking Like an i n t m p o l a t i o n phopehty,giwing t h e hesuet that i d a neah-king N (with nome addition& p k o p u d i u l 06 rmppingb on a ghoup r "intehpokk..tes" at zmo and at &a o t h m p k c u t h e n N "intmpakk..tudl &eady at a k b & w u j ( ~ . i n i t & q )many p o d ( 4 . 6 5 ) . m o , n u - h i n g b n i g k t be t h e appkophiate t o o l t o d e w h p a "non-ab&i.an homobgiCae algebha" ( 9 . 2 6 4 ) and dhow up again i n &geb&c topology ( 9 . 2 6 2 1 , 6unctiovd a d y b h ( 9 . 2 6 1 ) and i n
Nem-hing Zheohy h
Preface
xi
categohies xkth ghoup objeotn ( 9 . 2 6 5 ) . FinaUy, t h e a u t b h hopa that neathingo and " n m - k i n g m o d d a b t ( = N-ghOup6) Wiee pmue ;to be u&zdd doh Q n m b m 06 theohiG which .thy t o genehdeize "fineat" hedu&% t o t h e "YPOMf i n e a t cahe", doh . i n h t ~ n ~i en t h e t h e o k i a 06 automata and dynamicat bybtemb (bee § 9 ill, t o make t h e phovehb "16 you Xhy t o non-fineahize, you luiee dind t h e neah-hingb nice" come thue. Fmm t h e h i n g - t h e o h d c d point 06 view, many b i z m e bitua,t.bnts m i h e i n neah-hingb. f o h example, not evehy Le6.t i d e d .ih a hubneah-hing. Howeveh, thehe me bevehd bpLLcLte appUcaChm 06 nm-hingn 2O hi4 t h e o w (doh t h e fim-Rhtg-hehuk% dhow what h con&ined 2O hingn and whdt , .ih n o t ) and t o d v m d dgebha (becawe a h i g h pacentage 06 dedinitiom and ha& 06 t h e neah-hing theohy c a y oveh t o u n b ~ ddg e b h a ) . On quotations: Redehenced t o o t h m b e o t i o n h ahe done e.g. b y " Z d ) " meaning " 5 2 , beckLon d ) " oh by " 2 d 3 ) " abbhevhting "52, section d ) , numbeh 3 ) " , Numbm 6ok'lowing nama od authom hedeh t o t h e bibi!.iogmphy at t h e end 06 given, a.U p a p e a 06 t k i b authoh t h e book. 16 o n l y t h e authoh'b name c i t e d i n t h e bib&ogmphy ahe meant. T k i e bibfiagtraphy a b d d be ~ a i h a y cornplWe ah A a h ah nm-hingb ahe concehned. F O h nem-dieL& and hdaZed bubjectb we o n l y U t .thohe papeh?, which dihec;teY in@uenced t h e mat&L i n t k i b book. T k i n bibfioghaphy ~ a compiled 6 i n domm yea^ by J.C&y, G.B&ch, J.Mdone, H.UeatheMy and t h e authoh. Nama i n b u c k & hedeh t o t h e l?,ibt 06 "Supp&nwUa~y WOht2A11which c o n t a - h t h e non-neaA-h&g-papm oited tkio book. Sevehd hebu&h i n t k i n monaghaph ahe n w oh i n a new (and hope&Uy hnphoved) &hm UlithoLLt b p e c i d noLice. In t h e beginning 06 p o o 6 b thehe h no hepetition 06 t h e ahbump-tiom ( t o have space). l I * I t and l t e f I mean t h a t t h e diheotion indicated .ih h e a t e d at moment
(in p o o i b 06 equivdenced).
It 0 a pLmwle t o thank Utr. E . Fhedh.ibbon 06 t h e ed.Ltohid btadd 06 t h e Nohth-HuLlund Pubuhing Company and t h e heviweh doh a pLeaant cuopehhation and a L o t oh uhadul buggebXionh. Many thank4 go &a t o h. Hobpodah doh
xi i
Preface
he& excelLent typing j o b and t o G . ReLbch, Y.-S.
So, H.E. ReRC, J.D.P.
Meldhum and t o M . L . Holcombe d o h heading p a & 06 t h e manwchipt and ptouiding wed& kid and &po.ILtant commenL5. h(o4.t 06 CLee 1 haue t o thank my wide doh h a pcLtience and endutance i n fiuing w d h an aboevLt-mindedhunbund i n t h e p a t ye&. And now good Puck and much dun w L t h nem-hingb!
Remahh: A "Nedt~-h.CngNe~14leeMeh" c o m a ouA once oh twice a yeat, c o n t a i n i n g in@~ma-tiona b o u t t h e hecent deuelvpmed i n t h e theo.ty 04 n e m hingb. 16 you want t v o b t a i n c o p i e ~ ,w h i t e t o A. Obwaed o h t o t h e aLLthoh 06 tkin book.
FROM THE PREFACE TO THE FtRST EMTiON a "mhg ahe g e n w z e d hingd. Roughly dpoken, a nem-hing ( N , + , . I , &ehe + i.6 not nece.6bafdi.g a b d i a n and wikh o n t y one dinMblLtive
Nm-hhgd
bw,,.
Neah-hingd ahide i n a natuhae way: &tabe t h e be2 Mlr) 0 6 att M p p h ~ g d 06 a gmup (r,+l i n t o &e..t.ed, de6ine additian + point-whely and o a6 cornpabLthn. Then (M(r1 ,+, 0 ) i.6 a n m - h i n g . Even i 6 r h a b e l h n , o n l y one disthibutive &UI iA aeUayb 6utd.ieeed: ( i + g ) q h= 6oh+goh holds by t h e de6inhXon 06 6+g white doh 6o(g+h)= dog+deh we mutd have to adbwne t h a t 4 i.6 a homomohphibm. Anothm cxrunpte i.6 d U p p ! i d by t h e polynonciaed w . 8 . t . addLtion and hub6.titLLtion. A WeU-known tlebUet i n hing theotly dayb that e v a y hing can be embedded into t h e hing Elr) 06 a l l endomotlpL.bm 06 borne a b U n gmup r. Foh n m h g b we pmve ( 1.161 that e v a y nem-&g can be embedded into M(r) 60% dame gmup r. Hmce one migkt v i w hing theohy ab t h e " f i n m t h w h y 04 gmup mappingb", while n m - h i n g b pmvide t h e "non-.&npAIL thwty". SUILpILi6ingly, a t o t 06 ".&nm tle6uLtb" can be L t a n b 6 r n e d ~ t ot h e gmnehae c a e aiteh dLLitabte changes. Foh i n b m c e , t h e "aXomb1' 06 hing theohy, t h e phinritive h i n g d , m e d a d b e d by t h e ~auwusd e n b d y theohem 06 N . 3acobbon (doh h g s w i t h minimUn condition: W e d d m b m - M h t h w h m on bimpte h i n g d ) . foh nean-hiftgb bAni.kb tlebbcLetb cancaning phimit.iue neat-hingd wme 0 M a i n e d v i a t h e u m k 06 6QvQAd aLLthohd ( b u t t h e pmo$b aht to&d..ly d L 6 6 a ~ n t )t:h e hole a$ ffo%(V,Y) doh aingb iA played by M(r) oh dome hekzted t y p e s i n the n m - h h g m e (4.52, 4.54).
t d i s t O h i d y , t h e 6 h t 6tep tolllaha neah-hingb wab an a c O d C h e d Q W h done by V i c h o n i n 1905. ffe dhowed tfmt thehe do ex#t ' ' ~ i c t d bw i t h onty one d i s ~ % M v e W ' ( = n m - 6 i d d 6 ) . Some ylatch thebe nearr-dieeds & h o d up again and phDued t o be uae6LLe i n c o o k d i d z i n g cehtain impohtant daAbeA 06 g w m a c p . h e d ( h e m that lk6cahte'b method 06 coohdinatizing t h e " u b d " pkkne by t h e d i d d 06 heal numbm wa6 one 0 6 t h e m o d t ducceb6~u.t d t e p b .in geomeZq). It wa6 Zabbenhaus who wa6 abte t o d & W e all ~i.n..Lte n m - 6 i e C d ~ ( 1 . 3 4 ) . ~ U m d a y b ,n m - 4 i e l d b ahe a nrigkty tool h~chahactdz&g doubly thanbLtive gtloupd ( 8 . 4 4 1 , inincidence ghoups (8.6%) and Fmb&us gmupb ( 8 . 8 1 ) . Since t h e dwn 06 A3uo endomohppkibm 06 a n o n - a b m gmup ( r , + )i.6 not an endomotlpki6m i n genehat, t h e d e t d E ( r ) 06 aU 6 W e 4wn6 and di6?4menceb 06 endomohphh6 0 6 r m e consklehed. hebpect t o addition
Preface
X
and compobLtion, t h u e E ( r ) '6 ahe neah-hingb belonging t o t h e C h b 06 t h e "dib.thibU.t.iw&y gwnehated" neah-hingb Many p#u2 0 6 t h e we,U-ehtnblhhed fheohy 06 hingb wehe tRand5eNred t o n m hGzgd and n w neat-hing-bpecidic ~ e a t w uw4he dincovetled, bLLieding up a theoay 0 6 nu&-hingb b t e p b y b t e p . Up t o now, about 550 p a p m on Wah-hingb (and n u - d i d d s ) w i t h about 8000 pageb appeahed i n phint, b u t t h e m exha2 no book on t k i s b u b j e o t . Tkid book .thieb t o unidy t h e theohy and L t h feminology and t o g i v e a bgbtematic and w&-asbohted account 06 t h e phuent n t a t e 06 t h e t h e o q . Some aemahn am .to be made: ( a ) GenmaUy, 7 avoided t o g i v e ph006b doh t h e o h m which ahe &hm not &ng t h e main b . & m 06 dibcudb-ion oh a h long ~ ones which contain bpeciae mcthoh seemingly a,oflicable o d y i n .thLh context, cannot be b h p t 4 i e d b y pewhiow hunuets and invo.P,ve many o t h m ( e . g . geornethicat) dC?.tdied, b u t ahe heaiLiey accubib& in t h e f.Ltaatwre. ( b l Sevmak? hulLetd 6oUowing dhom uvLiwm& d g e b m oh &om t h e t h e o h y 0 4 ghoup with mubkple opnehatou ahe c i t e d , bu.t not p v e d i n a d m to dev,i.de hes1Letb which ahe bp&fiic doh n m - h i n g b and Rhobe which ahe
.
not. @L m y 6~wmbeing a m a e coUecLion 0 5 - t & w i d hesuets concaning dome "pathologiCae" b y b t m tU.ithout any application t o o t h m bzmches 06 mathematicn. A p a h t dotun t h e appfic&onb concaning axiomaticb and geomethy mentioned above, bpeciae & b b u 06 d i n i t e n u - h i n g b ( t h e AinLte "plans" nea-hingb) g i v e nw and highey eddi&nt C h b u 06 balanced p a h a w e A m ( 8 . 1 1 7 - 8 . 1 2 4 ) . Moheovm, incomplete block debigm aeheady [uith b& these p l m neah-hingb can be uaed t o chaaactmize Fhobg m u p , hence a h o 6.inite ghouph w i t h d i w e d - p o i n t - d u e a&tomohpkism ghoupb ( 8 . 9 6 , 8.971. 16 r i d a &inite, inwaAhaXy a h p l e n0n-abcL.h gmup, E ( r ) h "phimctive" and t h ee 6 o h e ewmy belb-map 06 r &ixing z m h .thk " b u m 0 6 endomohpkism" [exact @munLLeation i n 7 . 4 7 ) . Anotha v m i o n 06 t h e d e n b a y theohem 4.52 b h o u t h a t t h e d m a y p m p e h t y h ( i n t h e n e a - h i n g - c a e ) bometking Like an i n t m p o l a t i o n phopehty,giwing t h e hesuet that i d a neah-king N (with nome addition& p k o p u d i u l 06 rmppingb on a ghoup r "intehpokk..tes" at zmo and at &a o t h m p k c u t h e n N "intmpakk..tudl &eady at a k b & w u j ( ~ . i n i t & q )many p o d ( 4 . 6 5 ) . m o , n u - h i n g b n i g k t be t h e appkophiate t o o l t o d e w h p a "non-ab&i.an homobgiCae algebha" ( 9 . 2 6 4 ) and dhow up again i n &geb&c topology ( 9 . 2 6 2 1 , 6unctiovd a d y b h ( 9 . 2 6 1 ) and i n
Nem-hing Zheohy h
Preface
xi
categohies xkth ghoup objeotn ( 9 . 2 6 5 ) . FinaUy, t h e a u t b h hopa that neathingo and " n m - k i n g m o d d a b t ( = N-ghOup6) Wiee pmue ;to be u&zdd doh Q n m b m 06 theohiG which .thy t o genehdeize "fineat" hedu&% t o t h e "YPOMf i n e a t cahe", doh . i n h t ~ n ~i en t h e t h e o k i a 06 automata and dynamicat bybtemb (bee § 9 ill, t o make t h e phovehb "16 you Xhy t o non-fineahize, you luiee dind t h e neah-hingb nice" come thue. Fmm t h e h i n g - t h e o h d c d point 06 view, many b i z m e bitua,t.bnts m i h e i n neah-hingb. f o h example, not evehy Le6.t i d e d .ih a hubneah-hing. Howeveh, thehe me bevehd bpLLcLte appUcaChm 06 nm-hingn 2O hi4 t h e o w (doh t h e fim-Rhtg-hehuk% dhow what h con&ined 2O hingn and whdt , .ih n o t ) and t o d v m d dgebha (becawe a h i g h pacentage 06 dedinitiom and ha& 06 t h e neah-hing theohy c a y oveh t o u n b ~ ddg e b h a ) . On quotations: Redehenced t o o t h m b e o t i o n h ahe done e.g. b y " Z d ) " meaning " 5 2 , beckLon d ) " oh by " 2 d 3 ) " abbhevhting "52, section d ) , numbeh 3 ) " , Numbm 6ok'lowing nama od authom hedeh t o t h e bibi!.iogmphy at t h e end 06 given, a.U p a p e a 06 t k i b authoh t h e book. 16 o n l y t h e authoh'b name c i t e d i n t h e bib&ogmphy ahe meant. T k i e bibfiagtraphy a b d d be ~ a i h a y cornplWe ah A a h ah nm-hingb ahe concehned. F O h nem-dieL& and hdaZed bubjectb we o n l y U t .thohe papeh?, which dihec;teY in@uenced t h e mat&L i n t k i b book. T k i n bibfioghaphy ~ a compiled 6 i n domm yea^ by J.C&y, G.B&ch, J.Mdone, H.UeatheMy and t h e authoh. Nama i n b u c k & hedeh t o t h e l?,ibt 06 "Supp&nwUa~y WOht2A11which c o n t a - h t h e non-neaA-h&g-papm oited tkio book. Sevehd hebu&h i n t k i n monaghaph ahe n w oh i n a new (and hope&Uy hnphoved) &hm UlithoLLt b p e c i d noLice. In t h e beginning 06 p o o 6 b thehe h no hepetition 06 t h e ahbump-tiom ( t o have space). l I * I t and l t e f I mean t h a t t h e diheotion indicated .ih h e a t e d at moment
(in p o o i b 06 equivdenced).
It 0 a pLmwle t o thank Utr. E . Fhedh.ibbon 06 t h e ed.Ltohid btadd 06 t h e Nohth-HuLlund Pubuhing Company and t h e heviweh doh a pLeaant cuopehhation and a L o t oh uhadul buggebXionh. Many thank4 go &a t o h. Hobpodah doh
xi i
Preface
he& excelLent typing j o b and t o G . ReLbch, Y.-S.
So, H.E. ReRC, J.D.P.
Meldhum and t o M . L . Holcombe d o h heading p a & 06 t h e manwchipt and ptouiding wed& kid and &po.ILtant commenL5. h(o4.t 06 CLee 1 haue t o thank my wide doh h a pcLtience and endutance i n fiuing w d h an aboevLt-mindedhunbund i n t h e p a t ye&. And now good Puck and much dun w L t h nem-hingb!
Remahh: A "Nedt~-h.CngNe~14leeMeh" c o m a ouA once oh twice a yeat, c o n t a i n i n g in@~ma-tiona b o u t t h e hecent deuelvpmed i n t h e theo.ty 04 n e m hingb. 16 you want t v o b t a i n c o p i e ~ ,w h i t e t o A. Obwaed o h t o t h e aLLthoh 06 tkin book.