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Index of Subjects

A abelian group 2 A acts stably in G 108 A acts I-stably in G 108 additive group 2 algebra 232

group abgebra 232 linear 232 over a field 232

algebraic operation 1 autoDlorphisDl 5

inner 19 autoDlorphisDl group 21

outer 19 avoid

M avoids H/K 53

B Baer dispersive group (=Sylow tower group) 49 base group of wreath product 22 basis of free group 169 binary algebraic operation 1 Boolean lattice 228

C Carter subgroup 64 Cartisian product 13 central series

lower 39 upper 39

central chief factor 37

centralizer 11 ,24 center 12 characteristic siDlple group 20 characteristic subgroup 19 chief factor 25

central 37 I-central 97 .r-central/ .r-eccentric 74 Frattini 28 G-chief factor 25

chief series 25 class

extreDle 164 Fitting 78 of groups 58 priDlitive closed 69 ~saturated 69 U-closed 59 Lt-closed 120 saturated 69 Schunck class 69

1I"-closed group 36 closed operator 59 CODlDlutator 20 CODlDlutator subgroup 20 cODlpleDlent 33,205 cODlposition factor 25 cODlposition forDlation 161 cODlposition series 25 conjugate

eleDlents 11 subsets 11

core of a subgroup 27 coset

left 6

Index of Subjects

right 6 double 7

cover M covers HjK 53

covering subgroup X -covering subgroup

cyclic group 15

D Dedekind's identity 7 direct product 13 c,o-dispersive group 131

E epimorphism 5

63

equivalent of group pairs 22 exponent of group 15 exponent of formation 216 extension 23 external direct product 13

F factor 25

chief 25 composition 25 Frattini 28 normal 25 perspective 27 projective 27 .r-centre normal 74 I-centre chief 97 .r-eccentric normal 74

factor group 9 field 225 Fitting

class 78 subgroup 41

formation 60 composition 161 p-composition 161 generated by X 61 ll-critical 185 indecomposable 215 S-indecomposable 215 local 74 completely local 218

p-Iocal 74 7r-Iocal 74 n-tuple local 218 of classical type 194 primary formation 130 saturated 69 p-saturated 69 S-closed (=hereditary) Sn-closed 104 soluble saturated 161 S-formation 123 O-formation 175

formation function free group 169 Frattini

argument 28 extension 123 factor 28 subgroup 4

Frobenius Complement 151 group 151 kernel 151

96

.r-abnormal subgroup 156

.r-bijector 94

104

.r -central normal factor 74

.r -coradical 59

.r-critical maximal subgroup 84

.r -eccentric normal factor 74

.r-(/-) hypercentre 90,145

.r-limit maximal subgroup 93

.r-limit normal subgroup 84

.r-normalizer 84

.r-radical 59

.r-subnormal subgroup 156

G Gaschiitz subgroup 65 group 1

abelian 2 additive 2 Coo-group 50 cyclic 15 D.".-group 50 D~-group 50 E.".-group 50

253

254

E;-group 50 E~-group 50 finite 6 free group 169 Frabenius group 151 locally finite 173 operator group 20 nilpotent group 36 p-(-rr-) group 16,30 pd-group 30 rrd-group 30 rr-closed 36 rr-decomposable 70 p-normal group 56 p-( rr- ) nilpotent group 36 rr-special group 70 rr-split group 51 p-( rr-) soluble 30 rr-separable 44 p-( rr-) supersoluble 46 r-generated 164 primary 16 primitive 69 Schmidt group 51 Simple 8 Soluble 30 Supersoluble 46 Sylow tower group 49 Symmetric 2

group algebra 232

H H -conjugate 11 1l-defect 218 Hall subgroup 31 Hall rr-subgroup 31 Homomorphic image 65 homomorphism 5 hypercenter 90 /-(.1"-)hypercenter 90, 145

I ideal 229

left 229 right 229 two-sided 229

Index of Subjects

image of homomorphism 5 index of subgroup 6 injector

X-injector 68 inner automorphism 19 internal direct product 13 isomorphism 5

J Jacobson radical 235 Jordan-Holder theorem 26

K Kegel condition 127

L Lagrange's theorem 6 lattice 227

abstract 233 Boolean 228 complemented 228 distributive 228 of formations 178 modular 228

lattice variety 174 rr-Iength 44 linear algebra 232 local formation 74

1l1-critical 185 almost nilpotent 208

locally finite group 173 locally finite variety 173 lower central series 39

M maximal (G - H)-chain 156 maximal 8-subformation 175 maximal subgroup 4

.1"-critical 84 X-abnormal 65 X-normal 65

maximal inner screen 102 maximal screen 102 M covers HjK 52 M avoids HjK 52 minimal normal subgroup 25

Index of Subjects

minimal non-F-group 115 minimal screen 102 monomorphism 5 module 229

N

completely FpG-module 235 completely reducible 231 indecomposable 231 irreducible (or simple) 231 finitely generated 232 free R-module 232 over algebra 234 projective 235

natural homonorphism 9 nilpotent class 39 nilpotent group 36 nilpotent defect 219 normal subgroup 8

F-(I- )hypercentral 90,145 normal closure 40 normal factor 25

F -central 74 F-eccentric 74

normalizer 11

o operator of group 20 operator group 20 operation on class of groups 59 order

of element 15 of group 6 partial 226

outer outomorphism group 19

p

p-( 7r-)closed group 36 pd-( 7rd)-number 30 p-( 7r- ) group 30 p-normal group 56 p-( 7r-) nilpotent group 36 7r-special group 70 p-( 7r-)soluble group 30 7r-separated group 44 p- (7r- )supersoluble group 46

partial order 226 perspective factors 27 principal series 25 primary group 16 primitively closed class 69 primitive group 69 product

of classes of groups 61 direct 13 semidirect 21 subdirect 14 wreath 22

projective factor 27 projective module 235 projector

Ir-projector 67

R radical

F-radical 59 Jacobson radical 235

rank of free group 169 right coset 6 ring 228

S saturated

class 69 69

p-saturated class 69 Schmidt group 51 Schuncek class 69 Schur-Zassenhaus theorem 33 Screen 96

inner 99 minimal 102 maximal 102 maximal inner 102 full 118 S-closed 104 Sn-closed 104

semigroup 234 semidirect product series

21

chief( =principal) composition 25 normal 25

25

255

256

subnormal 25 set

partially ordered 226 ordered 226

Shemetkov-Baer function 161 Shemetkov formation 123 simple group 8 soele of group 42 soele of module 235 soluble group 30 subgroup 3

Carter subgroup 64 characteristic 19 commutator subgroup 20 X-covering 63 X-abnormal 65 .r-limit normal 84 Fitting subgroup 41 .r-subnormal 156 Frattini subgroup 4 Gaschiitz subgroup 65 Hall subgroup 31 Hallll"-subgroup 31 maximal 4 X-maximal 66 minimal normal 25 normal 8 subnormal 25 Sylow (p-)subgroup 17 X-abnormal 65 X -maximal 66 X-covering 63

Subnormal series 25 factor 25 subgroup 25

subring 229 subset 225 supersoluble group 46 supplement 70 Sylow

subgroup 17 system 95

Sylow's theorem 17,18 Sylow tower group 49 <p-Sylow tower group 131

Index of Subjects

symmetric group 2 system normalizer 95

U upper central series 39

V Variety of groups 169

indecomposable 215 locally finite 173

W wreath product 22

base group of 22

x X -covering subgroup 63 X-injector 68 X-projector 67

Z Zorn lemma 227

List of Symbols

H$G 3 l,,( G) 44

H<JG 8 G ::x:lA( or AD< G) 21,22

<M> 4 AlB 22 1\", 1\"', 1\"(n) , 1\"(G) 30

<a> 4 1\"(F) 59 [a,b] 20 O,,(G) 33 G'(= [G,G]) 20 On' (G) 44 [A,B] 39 ~p(G),~,,(G),~(G) 40,41 Z(G) 12 Ha 27 4>(G) 4 Ha 40

Imtp 5 GP 52

Kertp 5 Soc(G) 42 Soc(M) 235

G':::!.H 5 Rad(M) 235 HZ 11 Kn(G) 39 Ha 6 Zn(G) 39 aH 6 Z(G) 12 AxB 7 c(G) 39

IGI 6 </>,N,Np,N" 58

A1A2 ·• ·An 6 A, A( n), S, S" 58

G1 X G2 x··· X Gn 13 9", 9,U, £(or(l)) 58 G:F, G:F 59

IG:HI 6 SX,SnX,HX 59 G/H 9 DX,RoX,NoX 59 HcharG 19 EX, E4>X, EcI>pX 59 Na(H) 11 X1X2 61 Na(fl) 94 formX 61

Ga(H) 11 lform X 102

GA(G,H) 106 form G 61

Ga(H/K) 23 dorm X 161 Sform X 176

exp(G) 15 Oform X 175 AutG 19 X(p) 74 InnG 19 F(G) 85 OutG 19 Z~(G) 90

258

Z!,(G) 145 Zoo (G) 90 NG(n) 94 11"(.1") 59 CL(J) 161 LF(J) 96 hnh 98 N X 128 .1"1r 128 H <l <lG 156 H <l <lFG 156 .1"x 164 IX 164 X" 165 r 165 m(X) 165 F(X) 169 Fn(X) 169 Foo(X) 169 V(G) 169 < X~R > 171 varX 172 <P8(.1") 175 finM 178 :Fi 1\ .1"2 178 .1"1 V.1"2 178 11. VI M 203 Lo 203 LI 203 Ln 217 LI(.1") 206 n Ai 225 iEI

U Ai 225 iEI A ~ B 225 A c B 225 A\B 225 SuPAB 226 infAB 226 bla 227 .1"IIM 209 J(A) 235

List of Symbols