Publ. Ma . 49 (2005), 213–223
A CLASS OF GENERALIZED SUPERSOLUBLE
GROUPS
Adol o Balles e -Bolinches and Ta iana Ped aza
Abs ac
This pape is de o ed o he s udy o g oups Gin he uni e se c¯
L
o all adical locally ini e g oups wi h min-p o all p imes psuch
ha e e y δ-chie ac o o Gis ei he a cyclic g oup o p ime o de
o a quasicyclic g oup. We show ha wi hin he uni e se c¯
L his
class o g oups beha es e y much as he class o ini e supe soluble
g oups.
1. In oduc ion
A g oup Gis said o be supe soluble i i has a ini e no mal se ies o
cyclic ac o s. A chie ac o o a supe soluble g oup is cyclic o p ime
o de and a maximal subg oup has p ime index. In ac hese wo p op-
e ies cha ac e ize ini e supe soluble g oups. I is well known ha in
he uni e se o all ini e g oups he class o supe soluble g oups o ms a
subg oup-closed sa u a ed o ma ion which is in e media e be ween he
classes o nilpo en g oups and soluble g oups. Se e al au ho s ha e in-
es iga ed supe soluble g oups, no necessa ily ini e, and gene aliza ions
(hype cyclic g oups and locally supe soluble g oups) ex ending some e-
sul s o ini e supe soluble g oups and es ablishing connec ions be ween
hese classes o gene alized supe soluble g oups.
In his pape , we in oduce a class o gene alized supe soluble g oups
in he uni e se c¯
Lo all adical locally ini e g oups wi h min-p o all
p imes p: he class U∗. I is in e media e be ween he classes o supe -
soluble c¯
L-g oups and hype cyclic c¯
L-g oups and i plays a simila ole
in he class c¯
Las supe soluble g oups do in he class o all ini e g oups.
2000 Ma hema ics Subjec Classi ica ion. 20F16, 20F19, 20F50.
Key wo ds. Locally ini e g oups, gene alized supe soluble g oups, majo subg oups.
This esea ch is suppo ed by G an BFM-2001-1667-C03-03, MCyT (Spain) and
FEDER (Eu opean Union).
214 A. Balles e -Bolinches, T. Ped aza
2. P elimina ies
The main pu pose o his sec ion is o es ablish he no a ion, e -
minology and some esul s which will be used h oughou his pape .
No a ion ha is no speci ically ci ed he e is consis en wi h ha used
in [7], [8] and [10].
An in ini e g oup can ha e insu icien maximal subg oups o e en
none a all. In o de o a oid his si ua ion, Tomkinson [11] in oduces
he no ion o majo subg oup. We ecall his de ini ion. Le Ube a
subg oup o a g oup Gand conside he p ope ly ascending chains
U=U0< U1<···< Uα=G
om U o G, hen m(U) is he leas uppe bound o he ypes αo all
such chains. Clea ly m(U) = 1 i and only i Uis a maximal subg oup
o G.
A p ope subg oup Mo Gis said o be a majo subg oup o G
i m(U) = m(M) whene e M≤U < G. This is a nice ex ension o he
concep o maximal subg oup in he sense ha e e y p ope subg oup o
a g oup Gis always con ained in a majo subg oup o G[11, (2.3)]. In
pa icula , he in e sec ion o all majo subg oups o a g oup G, deno ed
by µ(G), is a p ope subg oup o Gwi h p ope ies simila o hose o
he F a ini subg oup o a ini e g oup.
In he sequel, we aci ly assume ha all g oups belong o he class c¯
L
o all adical locally ini e g oups wi h min-p o all p imes p.
Le Gbe a g oup and le Mbe a majo subg oup o G. Deno e
MG= Co eG(M). Then G/MGis ei he a ini e soluble p imi i e g oup,
i Mis a maximal subg oup o G, o a semip imi i e g oup, i Mis no
maximal in G(he e, a g oup Gis said o be semip imi i e i i is he spli
ex ension, G= [D]M, o a ai h ul di isibly i educible ZM-module D
by a ini e soluble g oup M). This esul , p o ed in [2], con i ms he
impo ance o majo subg oups in he s udy o he s uc u e o g oups
and mo i a es some de ini ions which a e in some sense ex ensions o
well known ones in he ini e uni e se.
De ini ion 1. Suppose ha Gis a g oup and le Mbe a majo subg oup
o G. We de ine
DM/MG=(Soc(G/MG),i Mis a maximal subg oup o G
(G/MG)0,i Mis no a maximal subg oup o G.
A Class o Gene alized Supe soluble G oups 215
In bo h cases, DM/MG=F(G/MG), (DM/MG)∩(M/MG) = 1 and
CG/MG(DM/MG) = DM/MG o e e y majo subg oup Mo G.
Le Gbe a g oup and conside wo no mal subg oups Hand Ko G
such ha Kis con ained in H. Then H/K is called a δ-chie ac o o G
i H/K is ei he a minimal no mal subg oup o G/K o a di isibly i e-
ducible ZG-module, ha is, H/K has no p ope in ini e G-in a ian sub-
g oups. E e y δ-chie ac o is ei he an elemen a y abelian ini e p-g oup
o some p ime po a di ec p oduc o ini ely many quasicyclic p-g oups
o some p ime p(see [7, (1.2.4)]).
Le Gbe a g oup. We say ha Gis a U∗-g oup i e e y δ-chie
ac o o Gis ei he a cyclic g oup o p ime o de o a quasicyclic g oup.
Ob iously he class U∗is a class o gene alized supe soluble g oups in
he uni e se c¯
Lbecause e e y ini e U∗-g oup is supe soluble and e e y
supe soluble c¯
L-g oup is ini e and so i is a U∗-g oup.
On he o he hand, since e e y g oup con ains minimal no mal sub-
g oups i ollows ha e e y U∗-g oup is hype cyclic (and hence locally
supe soluble [1]). Mo eo e he class U∗is in e media e be ween he
classes o supe soluble g oups and hype cyclic g oups. On one hand,
e e y quasicyclic p-g oup, pa p ime, is a non-supe soluble U∗-g oup.
On he o he hand, le Gbe a cyclic g oup o o de 22. Conside a
di isibly i educible ZG-module A, ai h ul o G, such ha Ais a pe i-
odic 2-g oup (Aalways exis s and i is unique up o isomo phism by [9,
(3.5)]). Then, applying [9, (5.9)], Ahas ank 2. In pa icula , he
Che niko g oup X=AG is no in he class U∗. Mo eo e , i Kis a
no mal subg oup o Xsuch ha X/K 6= 1 hen X/K con ains a min-
imal no mal subg oup N/K. Since X/K is a 2-g oup hen i is locally
nilpo en and hen, by [7, (1.2.6)], N/K is a cyclic g oup o o de 2. We
conclude ha Xis a hype cyclic g oup.
Le Bbe he class o all g oups in which e e y p ope subg oup has
a p ope no mal closu e. The class Bhas been in oduced and s udied
in [3], [4]. The esul s o hese pape s show ha his class is in e media e
be ween nilpo en and locally nilpo en g oups, and ha i is he na u al
gene aliza ion o he class o ini e nilpo en g oups om he ini e uni-
e se o he uni e se c¯
L. We show ha he la ges no mal B-subg oup
o a g oup is he Fi ing subg oup and e e y δ-chie ac o o a g oup G
in Bis cen al in G. Consequen ly, e e y δ-chie ac o o Ghas ank
one and Gis a U∗-g oup. The e o e, we ob ain in he uni e se c¯
Lsimila
inclusions o he ini e uni e se o hese classes o gene alized nilpo en
216 A. Balles e -Bolinches, T. Ped aza
g oups and gene alized supe soluble g oups:
{Abelian g oups} ⊂ {Nilpo en g oups} ⊂ B⊂ U∗
and all hese inclusions a e p ope .
Le pbe a p ime. We say ha a g oup Gis a Bp-g oup i Gis
p-nilpo en and he Sylow p-subg oups o Ga e nilpo en . The class Bp
is a local e sion o he class B. The esul s o he pape [5] show
ha Bpis a subg oup-closed o ma ion which plays he same ole in he
uni e se c¯
Las ini e p-nilpo en g oups do in he ini e one. In pa ic-
ula , e e y g oup Ghas a unique la ges no mal Bp-subg oup deno ed
by δp0p(G), o e e y p ime p, which is he in e sec ion o he cen alize s
o all δ-chie ac o s o Gwhich a e p-g oups and F(G) = Tpδp0p(G).
Le Gbe a U∗-g oup and le pbe a p ime. I H/K is a δ-chie ac o
o Gwhich is a p-g oup hen G/ CG(H/K) is a cyclic g oup o o de
di iding p−1 i p6= 2 o a cyclic g oup o o de di iding 2 i p= 2 by [7,
(1.5.18), (1.5.19)]. The e o e G/δp0p(G) is in he class A(p−1) i p6= 2
o in he class A(2) i p= 2, whe e A(n) deno es he class o all abelian
g oups wi h exponen di iding n. Mo eo e G/Op0p(G)∈A(p−1) o
e e y p ime pbecause Op0p(G) is he in e sec ion o he cen alize s o
all p-chie ac o s o Gby [7, (6.2.4)].
3. The esul s
A ini e g oup Gis supe soluble i and only i o e e y p ime p,
G/Op0p(G) is abelian o exponen di iding p−1. I Gis a c¯
L-g oup,
ha condi ion does no imply ha G∈ U∗(conside o ins ance he
example o a 2-g oup in Sec ion 2 which is no a U∗-g oup). Ou i s
esul p o ides a necessa y and su icien condi ion o a g oup G o be
aU∗-g oup.
Theo em 1. A g oup Gis a U∗-g oup i and only i o e e y p ime p,
G/δp0p(G)is in he class A(p−1) i p6= 2 o in he class A(2) i p= 2,
whe e A(n)deno es he class o all abelian g oups wi h exponen di id-
ing n.
P oo : Le Gbe a g oup such ha o e e y p ime p,G/δp0p(G) is in
he class A(p−1) i p6= 2 o in he class A(2) i p= 2. Conside H/K
a chie ac o o G. In pa icula H/K is a p-g oup o some p ime p.
Since G/ CG(H/K) is in he class A(p−1) i p6= 2 o in he class A(2)
i p= 2 i ollows om [8, B, (9.8)] ha H/K is a cyclic g oup o
o de p. Suppose now ha H/K is a di isibly i educible ZG-module.
In pa icula H/K is a p-g oup o some p ime p. Suppose ha p6= 2.
A Class o Gene alized Supe soluble G oups 217
Then L=G/ CG(H/K) is in he class A(p−1). Since H/K is a di isibly
i educible ZL-module which is ai h ul o Li ollows om [9, (3.1)]
ha Lis a cyclic g oup. Since |L|di ides p−1, i ollows om [9, (3.4)]
ha H/K has ank 1, ha is, H/K is a quasicyclic p-g oup. Assume now
ha p= 2. Then L=G/ CG(H/K) is in he class A(2). Applying [9,
(3.1)] we ha e ha Lis i ial o a cyclic g oup o o de 2. Consequen ly
H/K has ank 1 by [9, (3.4)]. We conclude ha Gis a U∗-g oup.
Ou nex esul s analyze he beha iou o U∗as a class o g oups
and hey a e mo i a ed by he well known ac ha , in he ini e uni-
e se, he class o all supe soluble g oups is a subg oup-closed sa u a ed
o ma ion [8, VII, (2.19)].
Recall ha a class Fo g oups is said o be a o ma ion i i sa is ies
he ollowing p ope ies:
1. I G∈Fand Nis a no mal subg oup o G, hen G/N ∈F.
2. I {Ni}i∈Iis a collec ion o no mal subg oups o Gsuch ha
G/Ni∈F o e e y i∈Iand Ti∈INi= 1, hen G∈F.
Theo em 2. The class U∗is a subg oup-closed o ma ion.
P oo : I is clea ha U∗is closed unde aking epimo phic images, ha
is, U∗is Q-closed.
Le {Ni}i∈Ibe a collec ion o no mal subg oups o a g oup Gsuch
ha G/Ni∈ U∗ o e e y i∈Iand Ti∈INi= 1. Since G/Ni∈ U∗
we ha e ha G/Ni/(δp0p(G/Ni)) is in he class A(p−1) i p6= 2 o
in he class A(2) i p= 2, o all i∈I. Le us deno e by GA(n)
he A(n)- esidual o a g oup G. Suppose i s ha p6= 2. Then
GA(p−1)Ni/Ni= (G/Ni)A(p−1) ≤δp0p(G/Ni) o all i∈I. In pa -
icula , GA(p−1)Ni/Niis in he class Bp o all i∈I. Since Bpis
a o ma ion by [5, Theo em 1] we ob ain ha GA(p−1) ∈Bp. Con-
sequen ly G/δp0p(G)∈A(p−1). Suppose now ha p= 2. Then
GA(2)Ni/Ni= (G/Ni)A(2) ≤δ202(G/Ni) o all i∈I. In pa icula ,
GA(2)Ni/Niis in he class B2 o all i∈I. The e o e GA(2) ∈B2
and hence G/δ202(G)∈A(2). I ollows om Theo em 1 ha G∈ U∗.
Consequen ly U∗is a o ma ion.
We p o e now ha U∗is subg oup-closed. Le Gbe a U∗-g oup and
le Hbe a subg oup o G. We ha e o p o e ha His also a U∗-g oup.
We spli he p oo in o h ee cases:
Case 1: Gis a Che niko g oup such ha G0( he adicable pa o G) is
a quasicyclic p-g oup o some p ime p. Since H0is a di isible subg oup
o G0, hen ei he H0= 1 o H0=G0. Fi s we assume ha H0=G0.
218 A. Balles e -Bolinches, T. Ped aza
Le A/B be a δ-chie ac o o H. Since he class U∗is Q-closed, we may
assume ha B= 1. I Ais a di isibly i educible ZH-module, hen Ais a
quasicyclic p-g oup. Suppose now ha Ais a minimal no mal subg oup
o H. Then ei he A≤H0o A∩H0= 1. I Ais con ained in H0,
hen Ais a cyclic g oup o p ime o de pbecause i is an elemen a y
abelian. Assume ha A∩H0= 1. Since AH0/H0is a minimal no mal
subg oup o H/H0and H/H0is a ini e supe soluble g oup, i ollows
ha AH0/H0is a cyclic g oup o p ime o de and so is A. Suppose now
ha H0= 1, ha is, His ini e. I A/B is a chie ac o o G, hen A/B
is a cyclic g oup o p ime o de . In pa icula (H∩A)/(H∩B) is
ei he i ial o a cyclic g oup o p ime o de . Hence a chie se ies o G
in e sec ed wi h Hwill yield, a e dele ing edundan e ms, a chie
se ies o H(o ini e leng h) wi h cyclic ac o s. The e o e H∈ U∗.
Case 2: Gis a Che niko g oup wi h Op0(G) = 1 o some p ime p. Then
G=G0M, whe e Mis a ini e subg oup o G. We can ce ainly assume
ha G06= 1, since o he wise Gis a ini e supe soluble g oup and so he
esul ollows. The e o e G0is a di isible abelian p-g oup o ini e ank.
By [12, (1.3)] he e is a ini e no mal subg oup Co Gcon ained in G0
such ha G0/C is a di ec p oduc o di isibly i educible ZG-modules,
say
G0/C = (G1/C)×(G2/C)× · · · × (Gn/C).
Since Gi/C is a δ-chie ac o o Gand G∈ U∗, we ha e ha Gi/C is a
quasicyclic p-g oup o all i∈ {1,...,n}. Le us i s assume ha C= 1
and deno e X1=G2×G3×· · ·×Gnand Xi=G1×· · ·× Gi−1×Gi+1 ×
· · · × Gn o all i > 1. I is clea ha Xiis no mal in G o all iand
Ti≥1Xi= 1. Mo eo e G/Xi∼
=GiMand (GiM)0=Giis a quasicyclic
p-g oup. Applying Case 1, we ha e ha H/(H∩Xi) is a U∗-g oup o
e e y i. Since U∗is a o ma ion, we conclude ha H∈ U∗. Assume now
ha C6= 1. By he abo e a gumen , HC/C is a U∗-g oup. We p o e
ha HC ∈ U∗. Le A/B be a δ-chie ac o o HC. The e is no loss o
gene ali y in assuming ha B= 1. Suppose ha Ais a minimal no mal
subg oup o HC. Then ei he A∩C= 1 o A∩C=A. I A∩C= 1,
i ollows ha AC/C is a minimal no mal subg oup o HC/C ∈ U∗and
so Ais a cyclic g oup o p ime o de . Assume ha A≤C. Le
1 = C0CC1CC2C···CCs=C
be pa o a chie se ies o Gpassing h ough C, ha is, Ci/Ci−1is a
minimal no mal subg oup o G/Ci−1 o all i∈ {1,...,s}. Since G∈ U∗,
we ha e ha Ci/Ci−1is a cyclic g oup o p ime o de o all i. On he
o he hand, since 1 6=A≤C he e exis s j∈ {1,...,s}such ha
A Class o Gene alized Supe soluble G oups 219
A≤Cjand ACj−1. In pa icula , 1 6=ACj−1/Cj−1≤Cj/Cj−1
and hen A∼
=ACj−1/Cj−1is a cyclic g oup o p ime o de . Suppose
now ha Ais a di isibly i educible Z(HC)-module. Then A∩Cis a
p ope ini e subg oup o A. This implies ha A/(A∩C)∼
=AC/C is a
di isibly i educible Z(HC/C)-module and so i is a quasicyclic g oup.
Consequen ly Ais a quasicyclic g oup. We ha e p o ed ha HC is a
U∗-g oup. Since U∗is a o ma ion and Cis an abelian no mal subg oup
o HC, i ollows om [2, Lemma 2] ha H∈ U∗, which is ou claim.
Gene al case: By [7, (2.5.13)], G/Op0(G) is a Che niko U∗-g oup o
e e y p ime p. Applying Case 2, we ha e ha H/(H∩Op0(G)) is a U∗-
g oup o e e y p ime p. Since Tp(H∩Op0(G)) = 1 and U∗is a o ma ion,
we conclude ha Hbelongs o U∗, which comple es he p oo .
Bea ing in mind ha he class o supe soluble g oups in he ini e
uni e se is sa u a ed, one can wonde i U∗could enjoy his p ope y in
ou uni e se. The answe is nega i e as he ollowing example shows:
Example 1. Le Xbe he egula w ea h p oduc o a quasicyclic
p-g oup and a cyclic g oup o o de p, whe e pis a p ime numbe . Con-
side G=X/ Z(X). Then G=G0Mis a semip imi i e g oup such ha
G0is isomo phic o a di ec p oduc o p−1 quasicyclic p-g oups and
Mis a cyclic g oup o o de p. Mo eo e , he g oup Gcan be exp essed
as G=Si≥1Gi, whe e Gi= Ωi(G0)M, o each na u al numbe iand
{Gi:i≥1}is an ascending chain o subg oups o G. No ice ha Gi
is a ini e p-g oup and hence a supe soluble g oup o each i≥1. How-
e e , i p6= 2, Gis no a U∗-g oup because G0is a di isibly i educible
ZG-module which is no quasicyclic. Applying [2, Theo em A], we ha e
ha U∗is no sa u a ed.
Ne e heless, U∗is closed unde aking ex ensions by Tomkinson’s
F a ini-like subg oup as ou nex esul shows.
Theo em 3. Le Nbe a no mal subg oup o a g oup Gsuch ha N/µ(G)
is a U∗-g oup. Then N∈ U∗.
P oo : Assume i s ha Gis a Che niko g oup. Then µ(G) is ini e by
[12, (1.2)]. On he o he hand, o e e y p ime p, (N/µ(G))/Op0p(N/µ(G))
is abelian o exponen di iding p−1 because N/µ(G)∈ U∗. Mo eo e ,
he a gumen s used in [11, (5.1)] allow us o show ha Op0p(N/µ(G)) =
Op0p(N)/µ(G). Thus, o e e y p ime p,N/Op0p(N) is abelian o expo-
nen di iding p−1.
Le H/K be a δ-chie ac o o N. Then ei he Hµ(G)/Kµ(G) is
N-isomo phic o H/K o Hµ(G) = Kµ(G) and H∩µ(G)/K ∩µ(G) is
220 A. Balles e -Bolinches, T. Ped aza
N-isomo phic o H/K. Consequen ly, he e is no loss o gene ali y in
assuming ha ei he K≤H≤µ(G) o µ(G)≤K≤H. In he i s case,
we ha e ha H/K is ini e and he e o e H/K is a p-chie ac o o N
o some p ime p. In pa icula , Au N(H/K)∼
=N/ CN(H/K) is ini e
and abelian o exponen di iding p−1. Applying [8, B, (9.8)], H/K is
a cyclic g oup o o de p. Assume now ha µ(G)≤K≤H. Then H/K
is isomo phic o a δ-chie ac o o N/µ(G)∈ U∗and consequen ly H/K
is a cyclic g oup o p ime o de o a quasicyclic g oup. We conclude
ha N∈ U∗.
In he gene al case, applying [7, (2.5.13)], G/Op0(G) is a Che ni-
ko g oup o e e y p ime p. Mo eo e , since N/µ(G)∈ U∗and
µ(G)Op0(G)/Op0(G)≤µ(G/Op0(G)) i ollows ha NOp0(G)/Op0(G)
is a no mal subg oup o G/Op0(G) sa is ying he hypo hesis o he he-
o em. By he abo e a gumen , we ob ain ha N/(N∩Op0(G)) is a
U∗-g oup, o e e y p ime p. Since U∗is a o ma ion, we conclude ha
Nbelongs o U∗, as equi ed.
Co olla y 1. Le Gbe a g oup. Then G/µ(G)∈U∗i and only i G∈U∗.
I Mis a majo subg oup o a g oup Gsuch ha DM/MGis a cyclic
g oup o p ime o de o a quasicyclic g oup hen G/MGis a U∗-g oup.
This ac mo i a es he ollowing:
De ini ion 2. Le Gbe a g oup and le Mbe a majo subg oup o G.
We de ine he ex ended index o Min G, deno ed by qG(M), as he ank
o DM/MG.
No e ha Mis a majo subg oup o G, hen qG(M)6= 1 i and only
i G/MGis no in U∗. Consequen ly he ollowing esul holds.
Co olla y 2. Le Gbe a g oup. Then G∈ U∗i and only i he ex ended
index o Mis 1 o e e y majo subg oup Mo G.
No e ha he abo e esul ex ends a well known esul o Huppe [8,
VII, (2.2)]. Ba hia [6] p o ed ha he in e sec ion o all maximal sub-
g oups o a ini e g oup o composi e index is a supe soluble cha ac e is ic
subg oup o he g oup. Ou aim now is o ob ain a simila esul in ou
uni e se.
De ini ion 3. Le Gbe a g oup. We de ine
L(G) = {M:Mis a majo subg oup o Gsuch ha qG(M)6= 1}
= {M:Mis a majo subg oup o Gsuch ha G/MG/∈ U∗}.
We s ipula e ha L(G) = Gi he abo e se o majo subg oups is
emp y.
A Class o Gene alized Supe soluble G oups 221
Theo em 4. Le Gbe a g oup. Then L(G)is a cha ac e is ic U∗-sub-
g oup o G.
P oo : I is clea ha L(G) is a cha ac e is ic subg oup o G. We p o e
now ha L(G) belongs o U∗. I he abo e se o majo subg oups is
emp y hen L(G) = G∈ U∗by Co olla y 1. Then we may assume
ha his se is non-emp y. Assume i s ha µ(G) = 1. Le L(G)U∗
be he U∗- esidual o L(G), ha is, he in e sec ion o all no mal sub-
g oups No L(G) such ha L(G)/N ∈ U∗. By Theo em 1, we ha e ha
L(G)/L(G)U∗is a U∗-g oup. We will show ha L(G)U∗is a subg oup
o µ(G). Ob iously L(G)U∗is con ained in e e y majo subg oup Mo G
such ha G/MG/∈ U∗. Suppose now ha Mis a majo subg oup o G
such ha G/MG∈ U∗. Then, he U∗- esidual o G,GU∗, is con ained
in MG. Since U∗is subg oup-closed, i ollows ha L(G)U∗≤GU∗.
Consequen ly, L(G)U∗is a subg oup o M. This implies ha L(G)U∗is
con ained in e e y majo subg oup o Gand hence L(G)U∗≤µ(G) = 1.
We conclude ha L(G)∈ U∗.
Suppose now ha µ(G)6= 1 and conside G/µ(G). By he abo e
a gumen we ha e ha L(G/µ(G)) ∈ U∗. Mo eo e , L(G/µ(G)) =
L(G)/µ(G). The e o e i ollows om Theo em 3 ha L(G) is a
U∗-g oup.
I is well known ha i Gis a ini e supe soluble g oup, hen Ghas a
no mal Sylow p0-subg oup [8, VII, (2.1)] o he smalles p ime pdi iding
he o de o G. Mo eo e , he de i ed subg oup o Gis nilpo en . The
co esponding e sions in ou uni e se o he class U∗a e he ollowing:
Theo em 5. Le Gbe a U∗-g oup. Then:
a) Ghas a no mal Sylow p0-subg oup o he smalles p ime pdi iding
he o de s o he elemen s o G.
b) G0≤F(G); in pa icula , G0belongs o B.
P oo :
a) Le Mbe a majo subg oup o Gand deno e MG= Co eG(M).
Suppose ha he esul is ue o G/MG o e e y majo sub-
g oup Mo G. Le Qbe a Sylow p0-subg oup o G. Since all p0-sub-
g oups o Ga e conjuga e and QMG/MGis a Sylow p0-subg oup
o G/MGby [7], i ollows ha G= (NG(Q))MG o e e y majo
subg oup Mo G. Since e e y p ope subg oup o Gis con ained
in a majo subg oup o G, i ollows ha G= NG(Q). Tha is,
Gcon ains a no mal Sylow p0-subg oup. Hence he e is no loss o
gene ali y in assuming ha MG= 1 and hen Gis ei he a ini e
