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