Publicacions
Ma emá iques,
Vol
36
(1992),
685-691
.
A
bs ac
THE
DETERMINATION
OF
ABELIAN
HALL
SUBGROUPS
BY
A
CONJUGACY
CLASS
STRUCTURE
WOLFGANG
KIMMERLE
AND
ROBERT
SANDLING
Pe e
Menal
in
memo iam
The
objec
o
he
a icle
is
o
show
ha
a
Jo dan-H&lde
class
s uc u e
o
a
ini e
g oup
de e mines
abelian Hall
subg oups
o
he
g oup
up
o
isomo phism
.
The
p oo
uses
he
classi ica ion
o
he
ini e
simple
g oups
.
A
conjugacy
class
s uc u e
on
a
g oup
cap u es in o ma ion
abou
i s
no mal
subse s,
o
example,
ha
which
is
deducible
om
i s
cha ac e
able
o
om
i s
in eg al
g oup
ing
.
Va ious
such
s uc u es
we e
in-
oduced
in
[KS]
and
used
he e
o
d aw
conclusions
abou
he
g oup
p e iously
in es iga ed
only
using
he
cha ac e able
o
he
in eg al
g oup
ing
.
The
mos
basic
class
s uc u e
conside ed
was
a
Jo dan-Hdlde
class
s uc u e,
one
which
cap u es
he
pose
o
no mal
subse s
o
a
g oup,
eco ds
hei
sizes,
indica es
which
no mal
subse s
a e
no mal
subg oups
and
which
a e he
p eimages
o
he
conjugacy
classes
o
i s
quo ien
g oups
.
I
was
shown
in
[KS]
ha
such
a
class
s uc u e
de e mines
he
chie
ac o s
o a
g oup
.
Mo e
echnically,
g oups
G
and
G*
ha e
he
same
chie
ac o s
i
hey
a e in class
co espondence
o
ype
JH,
ha
is, i
he e
is
a
bijec i e
co espondence
be ween
hem
p ese ing
no mal
subse s
and
hei
assumed
p ope ies
.
(Mo e
o mal
de ini ions
a e
a ailable
in
ou
ea lie
pape
.)
I
was
also
shown
ha
he
isomo phism
ype
o
an
abelian
Sylow
subg oup
is
de e mined
by a
class
s uc u e
o
ype
JH,
and
ha
o
an
abelian
Hall
subg oup
by
a
s onge
class
s uc u e
(namely,
one
in
which
he
se
o
p imes
in ol ed
in
he o de s
o
he
elemen s
in
conjugacy
classes
is
also
posi ed)
.
In
his
pape ,
his
ex a
hypo hesis
is
emo ed
o
gi e
he ollowing
.
68
6

W
.
KIMMERLE,
R
.
SANDLING
Main
Theo em
.
A
Jo dan-Hólde
class
s uc u e
de e mines
abelian
Hall
subg oups
up
o
isomo phism
.
I
should
be no ad
ha
he
esul s
o
ou
ea lie
pape
a e
dependen
on
he
classi ica ion
o
he
ini e
simple
g oups
.
Apa om
building
on
ou
ea lie esul s,
his
pape
makes
u he
in oca ions
o
he
classi ica-
ion
.
Ou
collabo a ion
on
he
opic
o
abelian
Sylow
and
Hall
subg oups
was
a
di ec
consequence
o
he second
au ho 's
pa icipa ion
[San]
in
he
1986
ing
heo y
con e ence
in
G anada
in
which
Pe e
Menal
played
a
p ominen
ole
.
We
begin
he
p oo
o
ou
heo em
by
ixing
some
no a ion
.
The
g oups
G
and
G*
will
be
assumed
o
be
in
class
co espondence
o
ype
JH
wi h
*
being
used
o
deno e
he
bijec ion
(on
subse s
and
subg oups
as
well
as
on
elemen s)
.
We
assume
ha ,
o
a
ixed
se
7
o
p imes,
G
has
abelian
Hall
7 -subg oups
.
We
mus show
ha
G*
also
has
abelian
Hall
7 -subg oups
and
ha hey
a e
isomo phic
o
hose
o
G
.
By
ou
esul
on
abelian
Sylow
subg oups,
i
su ices
o
show
ha
G*
has
Hall
7 -subg oups
and
ha hey
a e
abelian
(o
me ely
nilpo en )
.
The
p oo
o
he
de e mina ion
o
abelian
Hall
7 -subg oups
when
17 i
>_
2
is
mo e
elabo a e
han
ha
o
abelian
Sylow
subg oups
.
One
eason
o
his
is
he
absence
o a di ec
analogue
o
g oups
wi h
abelian
Hall
7 -subg oups
o
he
c i e ion
[KS,
2
.1]
o
de ec ing
abelian
Sy-
low
subg oups
.
This
can be
seen
om
he
g oup
ob ained
by
ex end-
ing
L2(7
5)
by
he
cyclic
g oup
C5
ac ing
as
ield
au omo phisms
(he e
7
=
{3,
5})
.
(Con as
he beha iou
o
Sylow
subg oups
as
seen
in
he
P oposi ion
below
wi h
ha
o
Hall
subg oups
.)
Unde
a
class
co espondence
o
ype
JH,
he o de s
o
x and
x*
need
no
coincide
( iz
.,
he
qua e nion
and
dihed al
g oups
o
o de
8)
.
This
ende s
subg oups
like
0"(G),
gene a ed
by
all
7 -elemen s
o
G,
less
use ul
han
hey
would be
i
o de s
we e
p ese ad
as
hey
a e
by
s onge
class
co espondences
(sea
[KS])
.
In hei place,
we
use
he
ollowing
cha ac e is ic
subg oup
o a
g oup
.
De ini ion
.
Le
Z
be a g oup and p a
se
o
p imas
.
De ine
C
p
(Z)
as
he
subg oup
gene a ed
by
all
x
in
Z
o
which
IZ
:
C
Z
(x)
j
is
a
p'-numbe ,
whe e CZ(x)
deno es
he
cen aliza
o
x
in
Z
.
Unde
a
class
co espondence
o
ype
JH,
Ci,(G*)
=
C,
(G)*
.
No e,
in
addi ion,
ha ,
i
G
has
abelian
Hall
7 -subg oups,
hen
hey
a e
all
con ained
in
C,
(G)
;
i
ollows
om a heo em
o
Wieland
[Suz,
5
.3
.2]
ha
0-'(G)
<
C,
(G)
.
In
he
p oo
o
Theo em
2
.1
o
[KS],
elemen s
no malising
each
simple
Lemma
1
.
Suppose
ha
ABELIAN
HALL
SUBGROUPS

68
7
ac o o
each
pe ec
minimal
no mal
subg oup
o
a
g oup
we e
exam-
ined
.
Fo
he
de e mina ion
o
abelian
Hall
subg oups,
such
elemen s
play
a
mo e
p ominen
ole
which
is
made
mo e
explici
in
he
ollowing
de ini ion
o
he
cha ac e is ic
subg oup
K(G)
which
hey comp ise
.
I
he
g oup
G
has
a
unique
minimal no mal
subg oup
and
his
subg oup
is
nonabelian
(so
ha
G
is
embedded
in
a w ea h
p oduc
[Rose,
p
.223]),
K(G)
is
he
in e sec ion
o
G
wi h
he
base
g oup
o
he
w ea h
p oduc
.
De ini ion
.
The
cha ac e is ic
subg oup
K(G)
o
G
is
de ined
as
he
in e sec ion
o
all
NG(S)
whe e
S
is
a
nonabelian
simple
subg oup
o
G
which
is
no mal
in
Soc
G,
he
socle o
G
.
Fo
a
nonabelian
minimal no mal subg oup
M
o
G,
le
K(GmodCG(M))
deno e he
in e se
image
in
G
o
K(G/CG(M))
.
I
is
easy
o see
ha
K(GmodCG(M))
is
he
in e sec ion
o
all
NG(S),
S
a
simple
subg oup
o
G
no mal
in
M
.
I
ollows
ha
K(G)
is
he
in e -
sec ion
o
all
such
K(GmodCG(M))
.
The
de ini ion
o
K(G)
is
simila
o
ha
o
McB ide's
k(G)
[McB,
p
.217],
and
he
wo
subg oups
coin-
cide
in
he
case,
impo an
he e
in
educ ion
a gumen s,
whe e
G
has
a
unique
and
nonabelian
minimal no mal subg oup
.
One
well-known consequence
o
he
classi ica ion
o
he
ini e
simple
g oups
which
was
used
in
[KS]
is
needed
again
he e,
and
is
s a ed
o
he
eade 's
con enience
(c
.
[GL,
7
.10])
.
P oposi ion
.
Le
S
be
a
simple
g oup
o o de
di isible
by a
p ime
p
.
I
X
is
a
subg oup
o
Au
S
in
which
S
<_
X
and
p
di ides
he
index
o
S
in
X,
hen
X
has
nonabelian
Sylow
p-subg oups
.
The
de e mina ion
o
abelian
Hall
subg oups
is
accomplished
h ough
a
se ies
o
educ ion
s eps
(c
.
he
p oo
o
he
main
heo em)
.
They
lead
o
he
ollowing
si ua ion
which
is
exploi ed
in
he
subsequen
lemmas
.
(i)
G
has
abelian
Hall
7 -subg oups
;
(ii)
each
minimal
no mal
subg oup
M
o
G
has
o de
di isible
by
some
p ime
in
7
;
(iii)
each
minimal
no mal
subg oup
M
o
G
is
nonabelian
.
Then
C
(G)
<
K(G)
and
C
(G*)
<
K(G*)
.
P oo
. .
Fo he
i s
conclusion,
i
su ices
o
show
ha
C

(G)
_<
K(GmodCG(M))
o
each
minimal no mal
subg oup
M
o
G
.
Le
x
be
an
elemen
o
G
o
which
IG
:
CG(x)j
is
a
7 '-numbe
.
By
(ii),
he e
is
some
pE
n
o
which
M
con ains
non i ial
Sylow
p-subg oups
.
Le
P
be
a Sylow
p-subg oup
o
G
con ained
in
CG(x)
.
Then
P
n
M
is
a
68
8

W
.
KIMMERLE,
R
.
SANDLING
non i ial
Sylow
p-subg oup
o
M
.
I
ollows
ha ,
i
S
is
a
simple sub-
g oup
o
G
no mal
in
M,
hen
x
ixes
a
non i ial
elemen
o
S
.
Thus,
xE
NG(S),
which
su ices
.
The
second
conclusion
ollows in
a
simila
way
using
he
esul
om
[KS]
ha
each
minimal no mal
subg oup
M*
o
G*
is
isomo phic
o
such
a
subg oup
o
G
(namely
M)
so
ha
pa s
(ii)
and
(iii)
apply
o
M*
.
While
Soc
G*
=
(Soc
G)
*
and
is
isomo phic
o
Soc
G
by
[KS]
in
his
case,
we
do
no
know
whe he
simila
s a emen s
hold
o
K(G)
( o
example,
unde
pa
(iii)
abo e)
.
We e
such
s a emen s
ue,
he
de e -
mina ion
o
abelian
Hall
subg oups
would be
accomplished
a his
poin
.
In hei s ead,
we
u n
o
a
close
examina ion
o
wo
ques ions
:
when
does
a
g oup
X,
S
<_
X
<_
Au
S
o
a
simple
g oup
S,
ha e
an
abelian
Hall
subg oup?
How
can
his
be
ecognised
using
p ope ies
de es able
unde
a
class
co espondence
o
ype
JH?
Suppose
ha
G
has
he
unique
minimal no mal
subg oup
M,
M
~
Sa,
o
he
nonabelian
simple
g oup
S
;
hen
G
may
be
iden i ied
wi h
a
sub-
g oup
o
Au
S'
which
is
a w ea h
p oduc
o
Au
S
and
he
symme ic
g oup
o
deg ee
a
.
As
ema ked,
K(G)
is
he
in e sec ion
o
G
wi h
he
base
g oup
(Au
S)a
.
In
ac ,
he e
is
a
subg oup
X
o
Au
S,
namely
he
image
o
NG(S)/CG(S),
o
which
K(G)
is
a subg oup
o
Xa
whose
p ojec ion
o
each
componen
is
su jec i e
.
The
nex
wo
lemmas
enable us o
de e mine
he
isomo phism
ype
o
an
abelian Hall
subg oup
o
Xa
on
he
basis o
in o ma ion
deducible
unde
a
class
co espondence
o
ype
JH
.
I
will
be
con enien
o
use
wo
i ems
o
s anda d
no a ion
in ol ing
a se
p
o
p mms
and a
g oup
Z
:
¡Zl
p
o
he
p
-pa
o
he
o de
o
Z
and
Z
p
o
a
Hall
p-subg oup
o
Z
.
The
i s
o
he
lemmas
makes
ano he
appeal
o
he
classi ica ion
o
ini e
simple
g oups
.
Lemma
2
.
Le
S
be
a
ini e
simple
g oup
and
le
Y
be
a
subg oup
o
Au
S
con aining
S
.
Le
k
be
a
di iso
o
¡Y¡
which
is
ela i ely
p ime
o
¡Si
.
Then
Y
has
a
unique
conjugacy
class
o
subg oups
o
o de
k
and
hese
subg oups
a e
cyclic
.
P oo
..
Fo
al e na ing
and
spo adic
g oups,
whose
ou e
au omo -
phism
g oups
a e 2-g oups, he
s a emen
is
acuous
.
Cyclic
simple
g oups
ha e
cyclic
ou e
au omo phism
g oups
so
he
lemma
is
s aigh -
o wa d
.
Fo
a
simple
g oup
S
o
Lie
ype,
only
a
ield
au omo phism
has
o de
ela i ely
p ime
o
151
(see,
o
example,
[MeB,
Lemma
4
.1])
.
Le
o,
be
he
se
o
p ime
di iso s
o
k
.

Now
Ou
S
has
a
cyclic
subg oup
ABELIAN
HALL
SUBGROUPS

68
9
(o
ield
au omo phisms) which
con ains
a cyclic
Hall
Q-subg oup
.
As
Ou
S
is
soluble,
Y/S
con ains
a cyclic
Hall
Q-subg oup,
and
so
Y
does
as
well
.
By
he
heo em
o
Wieland
ci ed
ea lie ,
all
Hall
-subg oups
o
Y
a e
conjuga e
and
e e y
subg oup
o
o de
k
is
con ained
in a
Hall
u-subg oup
so
ha
hese
subg oups
a e
also
conjuga e
in
Y
.
Lemma
3
.
Le
S
be
a
ini e
simple
g oup
.
Suppose
ha
X
and
Y
a e
subg oups
o
Au
S
con aining
S
and
ha
IX
I,
=
¡Y¡,
.
Then
X
has
abelian
Hall
7 -subg oups
i
and
only
i
Y
does
;
i
X,
and
Y,
a e
abelian
Hall
7 -subg oups
o
X
and
Y
espec i ely,
hen
X,
and
Y,
a e
conjuga e
in
Au
S
.
P oo
.
Le
u
=
7
-
7 (S)
and
le
k
=
IXI
o
=
¡Y¡,
.
Suppose
ha
X,
is
an
abelian Hall
7 -subg oup
o
X
.
By
he P oposi ion,
IX
:
Si
=
k
.
By
Lemma
2,
X,
= S~C
whe e S,
=
S
nX,
and whe e
C
is
cyclic
o
o de
k
.
Lemma
2 also
shows
ha
Y
has
a subg oup
D
o
o de
k and
ha
he e
is
an
au omo phism
a
E
Au
S
such ha
D
=Ca
.
I
ollows
ha
Xa~
is
a
Hall
7 -subg oup
o
Y
.
Tha
all
such
Hall
7 -subg oups
a e
conjuga e
in
Y
ollows
om
he
p e iously
ci ed
heo em
o
Wieland
.
Thus,
X,
and Y,
a e
conjuga e
in
Au
S
.
P oo
o
Main
Theo em
:
Recall
ha
ou objec
is
o
show
ha
G*
has
an
abelian
Hall
7 -subg oup
.
We
may
assume
ha
0,
,
(G)
=
1
=
O,
,
(G*)
.
No e
ha
0
7
"(G*)
=
0"'(G)*
;
as
0"'(G)
<
C,
(G),
0"
(G)
*
<_
C,,(G)*
=
C,,
(G*)
so
ha
C,
(G*)
con ains
all
7 -elemen s
o
G*
.
By
induc ion,
we
may
assume
ha
any
p ope
quo ien
o
G*
has
abelian
Hall
7 -subg oups
.
We
may
also
assume
ha
G
and
G*
a e
no
simple
by
[KS]
.
Le
M
be a minimal no mal subg oup
o
G
so ha
G*1M*
has
an
abelian
Hall
7 -subg oup
.
As
M*
~:d
M,
M*
also
has
an
abelian
Hall
7 -subg oup
.
By
he
heo em
o
Wieland
ci ed
abo e
and
a
heo em
o
Hall
[Suz,
5
.3
.12],
G*
has
a
Hall
7 -subg oup
.
By
ou
ea lie
Sylow
esul ,
each
Sylow
p-subg oup
o
pE
7
is
abelian
;
i
emains
o
show
ha
a
Hall
7 -subg oup
is i sel
abelian
.
As
i s
image
in
any
p ope
quo ien
o
G*
is
abelian,
we
may
assume
ha
M*
is
he
unique
minimal
no mal
subg oup
o
G*
and
hus
ha
M
is
he
same
in
G
.
I
M
and
M*
a e
abelian,
hen,
o
some
pE
7 ,
hey
a e
(elemen a y
abelian)
p-g oups
.
Fo
x* E
G*
such
ha
IG*
:
CG
*
(x*)j
is
7 ',
M*
<_
CG
*
(x*)
.
As
hese
elemen s gene a e
C,
(G*),
M*
cen alises
C,(G*)
whence
M*
commu es
wi h
all
7 -elemen s
o
G*
.
Thus
M*
is
a
cen al
subg oup
o
each
Hall
7 -subg oup
o
G*
.
Le
H*
be
a
Hall
7 -subg oup
o
G*
.
By
induc ion,
H*/M*
has
an
abelian
Hall
Q-subg oup
L*/M*
o
u
=
7
-
{p}
.
As
M*
is
o
cop ime

69
0

W
.
KIMMERLE,
R
.
SANDLING
index
in
L*,
i
has
a
complemen
K*
which
is
an
abelian
Hall
Q-subg oup
o
G*
.
Le
P*
be
a Sylow
p-subg oup
o
G*
con ained
in
H*
.
As
M*
<_
P*,
K*
no malises
P*
.
Mo eo e ,
K*
s abilises
he
no mal
se ies
P*
>M*
>_ 1
so
ha
K*
cen alises
P*
[Suz,
4
.1
.13]
.
Thus
H*
is
abelian
as
equi ed
.
Suppose,
hen,
ha
M
and
M*
a e
nonabelian
.
By
Lemma
1,
K(G)
con ains
all
Hall
7 -subg oups
o
G
.
As
desc ibed
abo e,
K(G)
is
iso-
mo phic
o
a
subg oup
o
X'
whe e
M
.z
;
S°,
S<
X
<_
Au
S,
K(G)
p ojec s
on o
each
di ec
ac o
X
and
X
has
abelian
Hall
7 -subg oups
.
In
a
simila
manne
K(G*)
is
isomo phic
o a
subg oup
o
Y°
whe e
M*
i
M
z ~
S°,
S
_<
Y
<_
Au
S
and
K(G*)
p ojec s
on o each
di ec
ac o
Y
.
By
Lemma
1,
C,
(G*)
_<
K(G*)
.
Bu C,(G*)
=
C,(G)*
and
IGI,
=
IC,(G)j,
so
ha
K(G*),
being
no mal,
con ains
a
Hall
7 -subg oup
o
G*
.
We
show
ha
i is
abelian
by
p o ing
ha
Y
has
abelian
Hall
7 -subg oups
.
Fo
his,
i
su ices
by
Lemma
3
o
show
ha
IXI71
=
IYL11
.
Fo
each
pEnn
7 (S),
X
and
Y
ha e
abelian
Sylow
p-subg oups,
in
he
la e
case
because
Y
is
he
image
o
K(G*)
which
has
abelian
Sylow
p-subg oups
.
By
he
P oposi ion,
X
p
and
Y
p
a e
con ained
in
S
so
ha
Ap
=
I
SI
p
=
Mp
.
I
emains
o
show
ha
IXI,
=
jY1,
o
u
=
7
-
7 (S)
.
By
Lemma
2,
X
Q
is
cyclic
so
ha
IXw
=
expX
Q
.
As
K(G)
_<
X'
and
as
K(G)
p ojec s
on o
X,
exp
X
Q
=
exp
K(G)
Q
.
Also,
exp
K(G)
o
=
exp
G
Q
=
exp(G/M)
asince
u
n
7 (M)
is
emp y
.
As
in
he
p e ious
case
G/M
and
G*/M*
ha e
isomo phic
abelian
Hall
-subg oups
so
ha
exp(G/M)
Q
=
exp(G*/M*)Q
.
As
be o e,
exp(G*/M*)
o
=
exp
G*
a
=
exp
K(G*)
Q
.
Finally,
Y
has
cyclic
Hall
o
,
-subg oups
by
Lemma
2
and
again
exp
K(G*)Q
=
expY,
=
¡Y
j,
.
T ac-
ing
h ough
he
equali ies,
we
conclude
ha
I
X
I
Q
=
¡Y¡,
as
equi ed
.
I
may
be
possible
o
ecognise
whe he a
g oup
has
nilpo en
Hall
subg oups
unde
a
class
co espondence
o
ype
JH
.
The
p oo
abo e
in
he
M
abelian
case,
o
an
easy
a gumen
using
he
Fi ing
subg oup,
p o ides
he
ollowing
esul
as
e iden e
.
P oposi ion
.
A
class
co espondence
o
ype
JH
de e mines
whe he
o
no
Hall
subg oups
o
a soluble
g oup
a e
nilpo en
.
ABELIAN
HALL
SUBGROUPS

69
1
Re e en es
[GL]
GORENSTEIN,
D
.
AND
LYONS,
R
.,
The
local
s uc u e
o
ini e
g oups
o
cha ac e is ic
2
ype,
Mem
.
Ame
.
Ma h
.
Soc
.
42,
no
.
276
(1983)
.
[KS]
KIMMERLE,
W
.
AND
SANDLING,
R
.,
G oup
heo e ic
and
g oup
ing
heo e ic
de e mina ion
o
ce ain
Sylow
and
Hall
subg oups
and
he
esolu ion
o a
ques ion
o
R
.
B aue ,
J
.
Algeb a
( o
appea )
.
[McB]
McBRIDE,
P
.,
Nonsol able
signalize
unc o s
on
ini e
g oups,
J
.
Algeb a
78
(1982),
215-238
.
[Rose]
RoSE,
J
.
S
.,
"A
cou se
on
g oup
heo y,"
Camb idge
Uni e si y
P ess,
Camb idge,
1978
.
[San]
SANDLING,
R
.,
"A
p oo
o,
he
class
sum
co espondence
us-
ing
he
eal
g oup
algeb a,"
Ring
heo y
(G anada
1986),
237-244,
Lec u e
No es
in
Ma h
.
1328,
Sp inge ,
Be lin,
1988
.
[SUZ]
SUZUKI,
M
.,
"G oup
heo y
II,"
Sp inge ,
New
Yo k,
1986
.
Wol gang
Kimme le
:

Robe Sandling
:
Ma h
.
Ins i u
B

Ma hema ics
Depa men
Uni e si i
S u ga

The
Uni e si y
7
S u ga
80

Manches e
GERMANY

M13
9PL
ENGLAND
Rebu
el
9
de
Gene
de
1992