Wreath products and fitting classes of C1-groups

Publicacions Ma emá iques, Vol 36 (1992), 205-215 . Abs ac WREATH PRODUCTS AND FITTING CLASSES OF C 1 -GROUPS J .C . BEIDLEMAN AND M .J . TOMKINSON A Fi ing class X o 61 - g oups is no mal i G is he unique X- injec o o G, o each G E 61 ; X is abelian no mal i GX > G' o each G E 61 . I is a well known esul o Blessenohl and Gaschü z ha he co esponding concep s coincide o ini e soluble g oups . He e we conside he w ea h p oduc p ope y (wpp) : X sa is ies wpp i whene e G E X and p is a p ime, he e is an in ege n such ha G' 2 C p E X . An abelian no mal Fi ing class sa is ies wpp bu a nonabelian no mal Fi ing class may no . Embedding heo ems ela ed e hose o Blessenohl and Gaschil z show u - he dis inc ions be ween abelian and nonabelian no mal Fi ing classes . Fo example, i X is an abelian no mal Fi ing class, hen s-T = 61 ; his is alse o nonabelian no mal Fi ing classes . 1 . In oduc ion In [1], we in oduced he concep o a Fi ing class X o i-g oups, o ce ain subclasses F o 6 1 , he class o soluble g oups in which each abelian sec ion has ini e o al ank . We ob ained a su icien condi ion o he exis en e and conjugacy o X-injec o s . Menegazzo and Newell [9] p o ed a o m o con e se so ha we ha e he ollowing esul : Le X be a Fi ing class o A-g oups . Then e e y -g oup has -T-injec o s i and only i , o each G E .lz, he e is a no mal subg oup M o G such ha (i) e e y Y-subg oup o G con aining he X- adical G_T is con ained in M and (ii) M/G is ini e . I X is such a .A-Fi ing class hen he 3 -injec o s o G a e necessa ily conjuga e . In [3], [4] i was obse ed ha in all he known examples, he subg oup M abo e can be chosen o be he 2j- adical o G o some no mal .A- Fi ing class 1-j . A A-Fi ing class 2,) is no mal i G-T is he unique 2j-injec o o G o each G E A . We say ha 2j is an abelian no mal .A-Fí ing class i G,_,j > G', o each G E A . I is a well known esul o 206  J.C . BEIDLEMAN, M .J . TOMKINSON Blessenohl and Gaschü z [5] ha o ini e soluble g oups e e y no mal Fi ing class is abelian no mal . This is no he case o 61-Fi ing classes . One o he mos impo an cons uc ions used in he ini e case o in es iga ing no mal Fi ing classes is he w ea h p oduc and ou aim he e is o see how his can be used in he in ini e case, whe e i is seen o be o mos alue in conside ing abelian no mal Fi ing classes and shows u he dis inc ions be ween he abelian and nonabelian no mal Fi ing classes . We say ha a A-Fi ing class X sa is ies he w ea h p oduc p ope y i , whene e G E X and p is a p ime, he e is a posi i e in ege n= n(G, p) such ha Gn 1 C E 3E, whe e Gn deno es he di ec p oduc o n copies o G . Fi ing classes o ini e soluble g oups wi h he w ea h p oduc p op- e y we e i s s udied by Makan [8] . He showed ha such classes a e no mal (and hence abelian no mal by Sa z 5 .3 o [5]) . Hauck [7] also in es iga ed Fi ing classes o ini e soluble g oups which sa is ied ce ain p ope ies o w ea h p oduc s . The w ea h p oduc p ope y seems o highligh some o he impo an ea u es o he p oo s o ou esul s and also illus a es mo e clea ly he me hods we a e using he e . Ou i s esul (Lemma 3 .1) shows ha an abelian no mal A-Fi ing class sa is ies he w ea h p oduc p ope y . This esul was one o he s eps in he p oo by Blessenohl and Gaschü z [5] ha a no mal Fi - ing class o ini e soluble g oups is abelian no mal . Thei p oo can be ex ended o no mal Fi ing classes o Ce niko g oups (Theo em 3 .6) . Howe e , examples show ha his esul does no hold o no mal A- Fi ing classes i he class F con ains nonpe iodic g oups e en i G/G .T is always ini e o i .A consis s o abelian-by- ini e g oups . The inal sec ion is de o ed o . embedding heo ems ela ed o u he esul s o Blessenohl and Gaschü z [5] . We show in Theo em 4 .2 ha i X is an abelian no mal A-Fi ing class hen O = l . This esul ails o nonabelian no mal Fi ing classes ; o example, we saw in [3] ha he class o Ce niko -by-nilpo en g oups is an s-closed Fi ing class o 651-g oups . The second embedding heo em o Blessenohl and Gaschü z [5] con- side ed he e is he embedding o ini e soluble g oups in G/G~ i X is a non-no mal Fi ing class o ini e soluble g oups . The esul s in he in ini e case a e a he mo e complica ed as he e a e ob iously many A-Fi ing classes such ha F-g oups can no be embedded in G/G X . The e a e A-Fi ing classes X such ha G/G~ is always ini e and, o any X which con ains all hype cen al A-g oups, G/G .T is ini ely gen- e a ed abelian-by- ini e [1, Theo em 3 .1] . Because o his we p o e wo FITTING CLASSESO 6 1 - GRouPS  207 esul s he e . Theo em 4 .5 asse s ha i X is a A-Fi ing class which is no abelian no mal and i H is a ini e soluble g oup hen he e is a A-g oup G such ha H is isomo phic o a subg oup o G/G~ . Ou second embedding esul (Theo em 4 .6) equi es he assump ion ha he e is a i-g oup L such ha L/L .T is in ini e nonabelian and some u he ai ly weak echnical es ic ions bu hen asse s ha i H is a ini ely gene a ed abelian-by- ini e g oup hen he e is a F-g oup G such ha H is isomo phic o a subg oup o G/G x . The p oo o Theo em 4 .5 is close o ha o he ini e case, again making conside able use o w ea h p oduc s . These esul s can be in e - p e ed as saying ha abelian no mal .A-Fi ing classes a e la ge (sx = .A) whe eas any o he Fi ing class is a he small since all ini e soluble g oups can appca abo e he X- adical . This p o ides a u he illus- a ion o he di 'e ence be ween abelian and nonabelian no mal Fi ing classes . These esul s also indica e some o he limi a ions o he w ea h p oduc in in es iga ing nonabelian no mal Fi ing classes . 2 . No a ion An C1-g oup is a g oup wi h a ini e no mal se ies whose ac o s a e abelian g oups o ini e ank and .whose o sion subg oups a e Ce niko g oups [9, Pa 2, p . 137] . The ollowing subclasses o C1 will be e e ed o he e . - ), he class o locally nilpo en (o hype cen al) C1-g oups '7 , he class o nilpo en C1-g oups 31, he class o ini e soluble g oups 11, he class o soluble Ce niko g oups (o ex emal g oups) 13, he class o polycyclic g oups 931, he class o soluble minimax g oups X -T = {G E 61 : G/GT E 9 _)}, whe e X and 1 -9 a e C1-Fi ing classes (-' : S7 i, he class o all Ce niko -by-nilpo en C1-g oups . Th oughou , A deno es an {s, Do}-closed subclass o 6 1 which is closed unde ini e soluble ex ensions . Examples o such A include 1, <_', 43, 9n and 6 1 i sel . Also he classes o abelian-by- ini e g oups in any o hese classes can also be aken o A . A F-Fi ing class :X is a subclass o J which is closed unde ascendan subg oups and such ha e e y .A-g oup which is a join o ascendan 3E-subg oups is an X-g oup . Le p be a p ime, wo o he less ob ious 20 8  J .C . BEIDLGMAN, M .J . TOMKINSON examples desc ibed in [1] ha will be e e ed o a e Q :(p) = {G E CH I : Soc,(G) < Z(G)} ~(p) = {G E 13 : G/CG(O,(G)) is a p-g oup} . We ecall ha a A-Fi ing class X is called a Locke class p o ided ha X* = X . P ope ies o Locke classes and he Locke *-cons uc ion a e gi en in de ail in [2] . I :X is a .A-Fi ing class he Locke sec ion o 9E consis s o all A-Fi ing classes SD such ha Qj* = X* . This is deno ed by Locksec(X) . 3 . W ea h P oduc P ope y Ou i s wo esul s p o ide examples o ce ain ypes o Fi ing class wi h he w ea h P oduc p ope y . Lemma 3 .1 . Le :X be an abelian no mal .A-Fi ing class . Then sa is ies he w ea h P oduc , p ope y . P oo : Le G E :3E and le p be ap ime . Le q be a p ime dis inc om p and le M be a ai h ul i educible Z,C, -module . Le Y= MC Q be he se nidi ec P oduc o M by C,, ; hen Y' = M . Le W = G Y and le B be he base g oup . Since A is closed unde ini e di ec p oduc s and ini o ex ensio is, W is a A-g oup . Also B E X and so W-e >_ B . Since X is abelian no mal, W-c _> BY' and so BY' E X . Le C be a subg oup o M o o de p . Then C a M and so BC a BY' and BC E X . Bu BC =G' 2 C Y , whe e n = IY : CI . The e o e, :X sa is ies he w ea h P oduc p ope y . Lemma 3 .2 . Le , :X be a, no mal L-Fi ing class . Le , G E X, le p be a p ime and Z a cyelic g oup o o de p . Then he e is a posi i e in ege m = m(G, p) such ha G` 1 Z E X o all posi i o in ege s n . In pa icula , :X sa is ies he w ea h P oduc p ope y . P oo : Assume ha he lemma is also . Since we a e only conside ing Ce niko g oups wc can choose a coun e example (G, p) such ha G is minimal ; ha is, he lemma holds 'o all p ope subg oups o G . By Lemma 3 .1 o [4], :X con ains all hype cen al (_ 11 -g oups . In pa icula Z E Y and so G :~ 1 . Thus G has a p ope no mal subg oup Gl such ha G/GI is a q-g oup o some p ime q . I q 7~ p, le Y be he g oup cons uc ed in Le nma 3 .1 wi h unique minin al no mal subg oup M o o de p' . I q = p, l Y = C q . In ei he case, le ZI be a subg oup o Y o o de q . FITTING CLASSFS OF 6 -GROUPS  209 Since G1 < G, he e is an in ege m such ha Gi " ? Z E X, o all posi i e in ege s n . Now o an a bi a y posi i e in ege n, conside W = Gnm1 ?Y . Le D = ( G" -1 ) Y be he base g oup o W and le D I = (G1 n ` 1 ) Y < D . As in he p oo o Lemma 5 .1 by Blessenohl and Gaschü z [5], D1 Z1 = G" P" l ZI E X and DZ /DI is a q-g oup . The e o e DI Z, is an ascendan subg oup o DZ and so DIZ1 <_ (DZ ),x . Bu D is a no mal .3C-subg oup o DZ, and so (DZi) . >_ DDIZ = DZ ; ha is, DZ, E X . I q = p, hen Gn,l 1 Z -DZ, E :X and we can ake m =ml . Suppose he e o e ha q 7~ p . In his case WT >_ D and W . is a maximal _X- subg oup o W . Bu D is p ope ly con ained in he 3E-subg oup DZ . The e o e D < W .T . Since W/D has a unique minimal no mal subg oup DM/D, we ha e DM <_ W . The e is a subg oup Z o M ha ing o de p and DZ < DM < W so ha DZ E :X . Bu DZ -G n,, j , " - 1 ` Z, Since ¡Y : Zi = p<" - 'q, and we can ake m = n a_I q . This comple es he p oo . The nex esul , which gene alizes a esul o J . Cossey [6, Lemma 2 .2], ndica es how adicals o Locke classes beha e in w ea h p oduc s . Lemma 3 .3 .  Le 3E be a A-Locke class and le G E .q :C . I H is a ini e soluble g oup and n a posi i e in ege , hen (G n 1H) . =B .- e , whe e B is he base g oup o Gn H . P oo .. Le W =Gn H ; hen he base g oup o W is B =Gn u `, whe e h = CHI . By Theo em 2 .9 o [2], B . = (G~),h . No e ha W/B . _~- (G/G . ) n H and, unde his iso no phism, B/B .T co esponda o he base g oup . The e o e, he cen alize o B/B . in W/B_T is con ained in B/B . . Bu [W-e, B] _< W n B = B . and so W cen alizes B/B . The e o e W . <B and so WT = B . -e, as equi ed . Theo em 3 .4 . Le 3 be a .C -Locke , class sa is ging i e w ea h p od- uc p ope Y . (i) I 2j E Locksec(X ), hen 1 -9 sa is ies he i ea h p oduc p ope g . (ii) 3C,~ = X (see Sa z 4 .1 o [7]) . P oo .- (i) This 'ollows in exac ly he same way as he esul is es ab- lished o ~-Locke classes in Lemma 5 .6 o [7] . (ii) Suppose ha :X~ =,¿ :Y and le G E )El .X . Then G/Gx is a ini e soluble g oup and hence con ains a subno mal subg oup H/G . o p ime o de p . Thus H .T =Gz and H/H .T = CP . By hypo hesis, he e is a posi i e in ege n such ha (H .T)n l C P E :X . Le W = Hn C P and le B = H'P be he base g oup o W . B,y Lem na 21 0  J .C . BI-1IDLGMAN, M .J . TOMKINSON 3 .3, WT = B .T . Bu W/B_ is a ini e p-g oup and so (H .T) n 2 C P sn W con a y o W- = Ba = (Hx)''' . We now conside he con e se o Lemma 3 .1 ; when is a Fi ing class wi h he w ea h p oduc p ope y an abelian no mal Fi ing class? Lemma 3 .5 . Le C_ 5)1 and le X be a q-Fi ing class such ha _D sí 1 .F . I :X sa is ies he w ea h p oduc p ope y hen X is an abelian no mal h-Fi ing class . P oo .. We show ha X* = F and hen i ollows om Theo em 2 .1 o [4] ha X is abelian no mal . Suppose hen ha X* :~ .IZ and le G E i X* . Then G/G x . con ains a subno mal subg oup H/G x . which is cyclic o p ime o de p . No e Hy . = G .. By Theo em 2 .3(d) o [2], H .T . /Hz is cen al in H and so H/HT is a ini e nilpo en g oup . Le P/H be he Sylow p-g oup o H/H . T . Then P/P . . =C P and P/Px is a ini e p-g oup . Since X sa is ies he w ea h p oduc p ope y, he e is a posi i e in ege n such ha (PT) n 2 C,, E X . Le W = Pn 1 C P and no e ha W/(Pz) - P is a p-g oup . Thus (PT)' 1 C P sn W and so (P : ) n 1 C P < W <_ W . . By Lemma 3 .3, WT . = (P . .)'P, which is a con adic ion . Hence X* = F , as equi ed . I should be no ed ha he condi ions on X and A in Lemma 3 .5 a e necessa y . Fi s ly, i is possible o ha e a Fi ing class :X sa is ying he w ea h p oduc p ope y bu no con aining all hype cen al F -g oups and in his case -T need no be a no mal Fi ing class . Fo example, we could ake X _  and q = ñ ~ . I we omi he condi ion ha . l C SI hen we can no say ha H/H . is ini e and he e will be no simila esul s . Fo example, le .A = T and :X = M ~- . Then X is s-closed and so is e en a Locke class . I G E M ~, hen G i C P E qT ~ and so '7 sa is ies he w ea h p oduc p ope y . Bu 911 is no ano mal %1-Fi ing class . The ollowing heo em shows ha a no mal (-"-Fi ing class is abelian no mal . This gene alizes Sa z 5 .3 o [5] . No e also ha i gene alizes Theo em 3 .2 o [7] . Theo em 3 .6 . Le , Y be an T-Fi ing class . Then he ollowing a e equi alen : (a) Y is a no mal, ( - - -Fi ing class . (b) :X sa is ies he w ea h p oduc p ope y . (c) X is an abelian no mal VE 1 -Fi ing class . P oo .. (a) implies (b) is Lemma 3 .2 . (b) implies (c) ollows om Lemma 3 .5 since ( _- C si 1 and any Fi ing class X which sa is ies he FITTING CLASSEES o 61-GROUPS  211 w ea h p oduc p ope y con ains all cyclic g oups o p ime o de and so con ains all hype cen al (- 11 -g oups . I is clea ha (c) implies (a) . One migh expec ha Theo em 3 .6 would ex end o u he classes .F o pe haps hold wi h ini eness condi ions on G/Gx . Howe e , i we ake A o be he class o abelian-by- ini e polycyclic g oups hen X= (EM n .A is a no mal F-Fi ing class by he main heo em o [3] and G/G X is ini e o each G E A . Bu G = C, > . 1 S3 has Gi~,n = Gol and G/G+n = S3 is nonabelian . 4 . Embedding heo ems This sec ion is de o ed o ob aining app op ia e gene aliza ions o Sa z 5 .3 and Sa z 6 .3 o [5] o Fi ing classes o 61-g oups . The ollowing simple lemma will be use ul in dealing wi h in ini e cyclic ac o s . Lemma 4 .1 . Le he g oup G be an ex ension o he g oup H by a a in ini e cyclic g oup (x) . I K = (H x H)((x,x-1)) < G x G, hen G is isomo phic o a subg oup o K . P oo .. The mapping 0 : G - K de ined by (hxn» = (hx',x- ") is a monomo phism . Theo em 4 .2 . I :X is an abelian no mal i-Fi ing class hen s :X = .ñ . P oo . Le G E .A ; hen G/G-T is abelian . By Lemma 3 .1 o [4], sj n .13 C X and so G/G~ is ini ely gene a ed [1, Theo em 3 .1] . We p o e by induc ion on he o sion- ee ank o G/G~ ha G E sX . Case 1 . = 0 . In his case G con ains an s :X-subg oup L = G x o ini e index and we use induc ion on ¡GIL¡ . Since GIL is ini e abelian i has a maximal no mal subg oup MIL such ha G/M is cyclic o p ime o de p . By induc ion, M E sX and so he e is an X-g oup X and a subg oup Xo o X such ha Xo - M . Now G is isomo phic o a subg oup G o o X o C p . By Lemma 3 .1 he e is a posí i e in ege n such ha X' 1 C p E :X . The e o e G= Go < Xo C p < X C p < X' i C p and so G E s-X . Case 2 . > 1 . In his case G has a no mal subg oup A such ha G/A is in ini e cyclic and A/G . has o sion- ee ank - 1 . Le T = ( ) be a cyclic g oup o o de 2 and le he w ea h p oduc W = G 2 T ha e base g oup B=G x G . Le G = A (x) and conside K = (A x A) ((x, x -1 ), ) <_ W . Then K3c > (G~ x G-T)K' . Bu , o each a E A, K' con ains he 21 2  J .C . BEIDLEMAN, M .J . TOMKINSON elemen (a, a -1 ) = [(1, a), ] .  Also K' con ains (x, x-1)2 = [( x -1 , x), ] and so K/K . has o sion- ee ank - 1 . By induc ion, K E sX . By Lemma 4 .1, G is isomo phic o a subg oup o (Ax A) ((x, x1)) <_ K and so GEsX . The ollowing wo lemmas, which a e used o es ablish Theo ems 4 .5 and 4 .6, a e gene aliza ions o Lemmms 5 .2 and 5 .3 o [5] . The p oo s o hese esul s a e he same as in [5] and hence a e omi ed . Lemma 4 .3 . Le X be a R-Fi ing class and le G E .A . Le G = N1N2 . . . N, ., whe e N i < G, 1 < i < . Then GX/11(Ni) . is con ained i= T in he cen e o G/ l(Ni) .T . i= Lemma 4 .4 . Le X and Y be g oups and le G = X Z Y . Le N be an abelian no mal subg oup o G which is con ained in he base g oup B o G . I Cc(B/N) is no con ained in B, hen X is abelian . Theo em 4 .5 . Le -Y be a A-Fi ing class which is no abelian no mal . I H E ~, hen he e is a i -g oup G such ha H is isomo phic o a subg oup o G/Gy . P oo . Since X is no an abelian no mal Fi ing class, he e is a g oup L E 1i such ha L/L_Y is nonabelian . Le H E ~ and le G= L 1 H . Le B be he base g oup o G so ha B= L', whe e m = CHI . Then B, < G .T and Bz/(Lx)' is abelian, by Le mna 4 .3 . Suppose ha G  is no con ained in 13 and le W = (L/L_ )1 H ; hen B /(L . )- is an abelian no mal subg oup o W . Since [G , B] < G x n B = B , i ollows ha G . /(L~) - cen alizes (L/L )~°/(B /(L )~`) . I ollows om Le nma 4 .4 ha L/L .y is abelian, con a y o ou choice o L . The e o e, G : < B and so G/G : > HG_ /G .T -- H . Le =X be any one o he ollowing 651-Fi ing classes : SI, 5) 2 , L 9Z, ~(p) o 5`) J (p) . Then :X is no abelian no mal, and Theo em 4 .5 shows ha i H is a ini e soluble g oup hen he e is an 6 1 -g oup G such ha G/G_ T con ains a subg oup isomo phic o H . I should be no ed ha his esul applies o nonabelian no mal Fi ing classes (e .g .  C, T) .  I X is one o he classes 1 2 , Q :(p) o s) ~ * (p) hen G/G  is always a ini e soluble g oup and so he e is no possibili y o embedding an a bi a y 651-g oup in G/GV . In o de e ex end Theo em 4 .5 he e o e i is necessa y o conside Fi ing classes such ha G/G .X is no always ini e . Mos o he A-Fi ing classes in which we a e in e es ed con ain 5) n .A ; his is FITTING CLASSCS O[' 6I-GRouPS  213 he case o no mal .A-Fi ing classes and, i A_D V, is ue o all .A-Fi ing classes such ha e e y A-g oup has 3E-injec o s . I X_D s) 1 .Fi hen G/Gx is ini ely gene a ed abelian-by- ini e and so we conside an ex ension o Theo em 4 .5 in which ini ely gene a ed abelian-by- ini e g oups a e embedded in G/Gx . Theo em 4 .6 . Le :X be a .A-Fi ing class such ha ) l A C_ :X and he e is a A-g oup L such ha L/Li is in ini e nonabelian (in pa icula , X is no an abelian no mal Fi ing class) . Suppose also ha X sa is ies one o he wo condi ions : (a) X* = 3E ; (b) he e is a .R-g oup T such ha T/T_ is in ini e nonabelian and has ini e cen e . I H is a polycyclic abelian-hy- ini e g oup, hen he e is a g oup G E .A such ha H is isomo phic o a subg oup o G1G_ . P oo :: The polycyclic abelian-by- ini e g oup H con ains a ee abelian no mal subg oup M o ini e ank , say, such ha H/M is ini e . We will show i s ha he e is a A-g oup B such ha B/Bx is nonabelian and B/Bx has a ee abelian no mal subg oup o ank a leas . Le L be a .F-g oup such ha L/LX is in ini e and nonabelian ; hen by Theo em 3 .1 o [1], L/Lx has a ee abelian no mal subg oup o ank s, say . Choose a posi i e in ege m such ha ms> and le B = Ln` . I X sa is ies condi ion (a) :X* = :X, hen Bx = (L,,) n ` [2, Theo e i 2 .9] and so B/Bx has a ee abelian no mal subg oup o ank ms >_ . Also B/Bx is nonabelian . So suppose ha X sa is ies condi ion (b) ; hen we may suppose ha L/Lx has ini e cen e . The e o e Z(B/(L .X)n`) (Z(L/Lx))' is also ini e . B,y Lemma 4 .3, Bx/(L .T)n` < Z(B/(Lx)`) and so B /(L . ) m is ini e . Now B/(Lx)' has a ee abelian no - mal subg oup A/(Lx)' o ank ms > . Since B . /(Lx) n ` is ini e, A/(Lx) " -- ABx/Bx and so ABx/Bx is a ee abelian no mal subg oup o B/Bx o ank a leas . Also, Since L/Lx is nonabelian, B/B .T mus be nonabelian . The e o e, using ei he (a) o (b), we ha e shown ha he e is a A- g oup B such ha B/B .T is ionabelian and has a ee abelian no mal subg oup A/B o ank a leas . Now le F = H/M ha e o de n and le G=B 1 F .  As in he p oo o Theo em 4 .5 we ha e (B_T)" <_ Gx <_ B n, he base g oup o G . Thus Gx = (B n )x . In case (a), (B ) n = (B n )x = Gx while in case (b), he same a gumen as o Bx/(L_T)' shows ha (B n)x/(B .T)n is ini e . Now A n /(B :Y) n is a ee abelian no mal subg oup o G/(B x )n