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8/18/2019 Theorems Like Sylow's (Hall)
1/19
THEOREMS LIKE SYLOW'S
By
P.
HALL
[Received 24 June 1955.—Read 17 November 1955]
1. Discussion
of
results
1.1. Notation.
Let
w
be any set of
primes
and let
w'
be the
comple-
mentary
set
consisting
of all
primes
not in
w. Ev ery positive integer
m
can be expressed uniquely in the form
m
=
m
m
m
VT
,,
where m
m
is the largest divisor of m which has no prime divisor in m'.
We denote the order of a finite group G by (G). If (6%, = (G ) we call
G a w-group. If
H
is a subgroup of G such tha t
(H) = (G)^,
we call
H
an
S
m
-subgroup of G. In
particular, when w consists
of a
single prime p,
an
$p-subgroup
is the
same
as a
Sylow ^-subgroup.
But for
general
-nr, a
Sylow m-subgroup
is
not the same as an ^ - sub grou p , bu t
is
an $p-subgroup
for some prime ^in-nr.
Let E
m)
C
m
, and
D
m
be the
following propositions about
a
finite group G.
E
m
: G has at least one ^ - s u b g r o u p .
C
m
: G satisfies E
m
and any two /S^-subgroups of G are conjugate in G.
D
m
: G satisfies C
m
and every -nr-subgroup of G is contained in some
#
OT
-subgroup
of
G.
Obviously D ̂can also be expressed in either of the alternative forms:
(i) G satisfies C
m
and
every maximal -nr-subgroup
of G is an
#
OT
-subgroup
of G;
(ii)
G satisfies E
m
and, if
H
is an
^ - s u b g r o u p
of
G
and
L
is any
xir-subgroup of G, then
L
^
H
x
for some
x e G.
1.2. D-theorems. A sufficient condition for a finite group to satisfy
D
w
m ay be called a D-theorem. Similarly for ^-theorems and C-theorems.
Our main object is to prove a new D-theorem, viz. Theorem D5, below.
The two basic D-theorems are, of course, those due to Sylow and Schur,
viz.
THEOREM
Dl. If p is a prime, then every finite group sa tisfies D
p
.
THEOREM
D2. / / G has a n ormal Abelian 8
m
.-subgroup, then G satisfies D
m
.
We mak e once
and for all the
obvious remark th at,
if
m
x
and
m
2
are two
sets
of
primes such that (G)
mi
— (G)^, then
for
this particular group G
the propositions D
mi
and J)
m%
are equivalent: G satisfies bo th or neither.
Similarly for the corresponding E-
and
C-propositions. So Sylow
3
s theorem
Proc. London Math. Soc. 3) 6 1956)
8/18/2019 Theorems Like Sylow's (Hall)
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THE OR EM S LI K E SYLO W'S 287
may be expressed by saying that G satisfies D
m
whenever at most one
prime in
m
divides
(G).
All -nr-groups satisfy D
m
trivially.
In an earlier paper (9), we proved
THEOREM
D 3.
If G is soluble, then G satisfies D
m
for all
m.
Quite recently, W ielandt (14) has derived a remark able new D -theorem,
which resembles Sylow's theorem and differs from D2 and D3 in the fact
that it does not reduce to a triviality for simple groups.
We define the propositions E%,
E^, 0%,
and
D
s
m
abo ut a finite grou p
G
as follows.
E^\ G has a ni lpotent ^- su bg ro up .
E% f\ G has a soluble ^- su bg ro up .
C
s
m
: G satisfies C
m
and its /S^-subgroups are soluble.
D ^ : G satisfies D^ and its -nr-subgroups are soluble.
Wielandt's theorem is
THEOREM D4. E% implies D
m
.
This makes it superfluous to introduce the propositions C^. and D
7
^,
analogous to
C
s
m
and
D%
but with 'soluble' replaced by 'nilpotent', since
they would both be equivalent to
E^.
1.3. That E% . does not in general imply D
m
or even C
m
is shown, as h as
been remarked before, by the simple group of order 168 which has two
distinct classes of conjugate o ctahed ral subgroups. This grou p satisfies
Uf.3 but not O
2
,3-
Wielandt suggests as a possibility that the presence of a supersoluble
^-subgroup might be sufficient to imply
C
m
or perhaps even
D
m
.
B u t
this is not the case, as is shown by the following examples.
We denote by
L
p
the simple group of order
%p(p
2
—l),
where
p
is a prim e
greater tha n 3. Then L
lly
of order 660, has two distinct classes of con jugate
$
2 3
-subgroups, one class being dihedral (and therefore supersoluble) and
the other class tetrah edra l. Again, L
ei
has two distinct classes of conjugate
$
2 3 5
-subgroups, one class being dihedral and the other icosahedral (and
therefore not even soluble). For these properties of L
p
, see Burnside (1),
Chapter XX.
However, it is easy to see that any two supersoluble S
m
-subgroups must
be conjugate. A more general result m ay be described as follows. Le t
p
x
, p
2
,..., p
r
be distinct primes. We say th at a finite group H has a Sylow
series of complexion (p
lt
p
2
,---, p
r
) if (H ) is divisible by no primes other
than p
1}
p
2
, -, p
r
and if, for each i = 1,
2,...,
r—1 H has a normal (and
therefore characteristic)
8
PltPt
^-subgroup. Then we may state :
8/18/2019 Theorems Like Sylow's (Hall)
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288 P. H A L L
T H E O R E M
A l .
Let p
v
p^,..., P
r
be the distinct primes which divide (G)
m
,
arranged in any given order. Then any two S
m
-subgroups of G both of w hich
have a Sylow-series of complexion (p
v
p
2
,--, p
r
) must be conjugate in 0.
The conjugacy of supersoluble /S^-subgroups follows from this theorem,
because a supersoluble group of order pi
1
p%
i
...p%
r
, where p
x
> p
2
> ... > p
r
,
always has a Sylow series of complexion (Pi,P2,---,p
r
)'
The group
L
n
does no t con tradict Theorem
A1
because its dihedral
#
2
g-subgroups h ave Sylow series of complexion (3,2) bu t none of com-
plexion (2,3), whereas its tetrahedral $
2)3
-subgroups have Sylow series of
complexion (2,3) but none of complexion (3,2).
I know of no counter-example to the conjecture that, if 2 does not
belong to m, then E
m
implies D
m
.
1.4. The principal theorem of the present pape r is
THEOREM D5 . / /
K is a normal subgroup of 0 such that K satisfies Ei^
and G/K satisfies D^, then G satisfies D% .
Here we merely note a few corollaries of this theorem.
A chain of subgroups
G = G
o
>
G
t
> ... >
G
r
=
1 of a group
G
will
be called a series of G if each term G
i
is norm al in the p receding te rm G^
v
thoug h n ot necessarily norm al in
G.
C OR OL L AR Y
D 5 . 1 . / /
G has a series G = G
Q
>
G
x
> ... >
G
r
=
1
such
that each of the factor groups
G
i
_
x
jG
i
satisfies E
1
^, then G satisfies D^.
It would be sufficient here to postulate that the first factor group of the
series,
G/G
1}
satisfies the weaker condition
D%
instead of
E%.
Following Cunihin (2), we say that a finite group is vr-separable if all
its c.f. (= composition factors) are Ttr'p-groups for various primes
p
in
m.
Thus G is tn--separable if and only if no two distinct primes in m divide
the order of any c.f. of G. If p is a prime, all finite grqmps are ^-separable
If G is both -nr-separable and xo-'-separable, then G is soluble: for th en the
order of any c.f. of G must have the form p
a
qP with p in TO- and qin m'\
but by a classical theorem of Burnside, groups of order p^qP are always
soluble, so th a t either a = 1, fi = 0 or else a = 0, £ = 1.
Clearly, if m
x
is any subset of m, then every m-separable group is also
•ro-j-separable. In (2), Theorem X I, Cunihin proves th a t all xtr-separable
groups satisfy C^, and therefore, by the preceding remark, also C
s
mx
for
any subset TD-J of m. This result is contained in
C OR OL L AR Y D 5 . 2 . All -m-separable groups satisfy
D
s
mi
,
for any subset
TO^
o
m
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THEO REM S LIK E SYLOW 'S 289
This corollary also contains as a special case the main Z)-theorem of
Cunihin's paper, Theorem XII, (1) and (2).
We call a finite group
G m-serial
if every c.f. of
G
is either a -nr-group
or else a -ur'-group. Equivalently, G is -nr-serial if it has a m-series, i.e. a
series G = G
o
> G
x
> ... > G
r
= 1 such tha t each of the factor groups
G
i
_
1
IG
i
is eithe r a xtr-group or else a -nr'-group. Clearly, -nr-serial groups
are the same as m'-serial groups. If p is any prime in w and q any prime
in xtr', then every xtr-serial group is p, ^-separable. Hence we have
COROLLARY D5.3. All m-serial groups satisfy D ,
pq
for any p in m and
any a in m'.
We note that, by the theorem of Burnside already mentioned, D
p
q
is
equivalent to D
s
vq
.
1.5. In his well-known textbook (15), Chapter IV, Satz 27, Zassenhaus
proves two C-theorems. The corresponding D-theorem s, which follow a t
once,
are
THEOREM
D6. / / G has a normal
8^-subgroup K such that G/K is soluble,
then G satisfies D
m
.
THEOREM D7. If G has a normal
S
m
>-subgroup
K such that K is soluble,
then G satisfies D
m
.
We call a finite group G vx-soluble if it is both -nr-separable and xo--serial.
Equivalently, G is xtr-soluble if every c.f. of G is either a ^p-group (and
therefore cyclic of order p) for some p in vx, or else a xcr'-group.t
In (3), Cunihin proves the following generalizations of D6 and D7.
THEOREM D6.* Every m-soluble group satisfies D^.
1
for any subset m
x
o f -m .
THEOREM
D7.*
Every m-soluble group satisfies D^Jor any subset TJT
X
of m .
Of these resu lts, Theorem D6* is contained as a special case in Corollary
D5.2.
Fo r the sake of completeness we include in section 2.5 th e deduction
of Theo rem
D7*
from Sch ur's Theorem D 2. Both D6* and
D 7*
are generali-
zations of Theorem D 3.
1.6. The theory of systems of permutable Sylow subgroups, developed
in (10) for the case of soluble groups, extends at once in an appropriate
form to all groups satisfying C
s
m
. This has been rem arked for th e case of
m-separable groups by Gol'berg (8).
t This terminology seems preferable to that of Hall and Higman (12). In that
paper, nr-serial groups as defined above were called -nr-soluble. B ut th e m -subgroups
of m -serial groups need no t always be soluble, whereas all m-su bgroups of a itr-separ-
able group are soluble, and in particular, all m-subgroups of a m-soluble group are
soluble. Owing to Burnside's theorem, no distinction arises between -nr-serial and
-nr-soluble groups as denned here unless -nr conta ins at least three prim es. I hope
that the present choice of terms is reasonably consistent with Cunihin's.
5388.3.6 XJ
8/18/2019 Theorems Like Sylow's (Hall)
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290
P . HA LL
A set of Sylow subgroups
P
i
of the finite group
G,
one for each p rime
p
i
in
w,
which are permutable in pairs (i.e. such that
P
t
Pj = P
i
P
i
for all
i, j),
will be called a Sylow m-system of G. Obviously, only a finite number of
the Pi can differ from 1. Owing to the pe rm utab ility of the P
i}
their pro duct
is an ^-su bg ro up of G and it is shown in (10), § 2, th at this product m ust
be soluble. Conversely, a soluble iS^-subgroup H of G has a Sylow m-system
(whose product is therefore H itself) and this will also be necessarily a
Sylow -nr-system of G as defined above.
In (10), § 4, it was shown that any two Sylow -nr-systems (...,
P
i}
...)
and
(...,
Qi,...)
of a soluble -nr-group
H
are conjugate in
H,
in the obvious sense
that there is an element
x
in
H,
independent of
i,
such that
Q
t
= P\
for
all i. Thus we may state at once:
T H E O R E M A2. G has at least one Sylow m-system if and on ly if it satisfies
E^. G has one and only one class of conjugate Sylow m-systems if and only
if it satisfies C^.
Theorem
D 3 allows the following conclusions to be draw n.
(i)
E% implies E^ for an y subs et m
1
of to-.
(ii)
D^ implies D*
ni
for any subset m
1
of TO-.
(i)
is immediate. As for (ii), let
G
satisfy
D
s
m
and let
L
be a maximal
xo-j-subgroup
of G. Then L is contained in. some /S^-subgroup H of G.
B ut
H is soluble. Hen ce, by Th eorem D 3, L is an
jS^-subgroup
of H and
therefore
also of
G.
S ince t he ^ - s u bg ro u ps of
H
are all conjugate in
H
and
t h e ^ - s u b g r o u p s H of G are all conjugate in G, it follows th a t G has
only
a single class of conjugate m axim al
TOi-subgroups
and th at these are
soluble iS^-subgroups of G. Thus & in fa ct satisfies Z )^ .
Cunihin
(2) defines a finite gro up to be
m-Sylow-regular
if it satisfies
C^
JTl
for
every subset
m
1
of
m.
Thus we conclude th at
all groups satisfying D^
are m -Sylow-regular; an d further, from Th eorem
A2,
all such groups possess
a single class of conjugate Sylow m
x
-systems for any given subset w
1
of
xtr.
In particular, this is the case for all groups G satisfying the hy pothesis of
Theorem D5 or Corollary D5.1. This statement is a generalization of
Gol'berg's result about -nr-separable groups. It is obvious that in a
itr-Sylow-regular grou p, the Sylow T&i-systems are all to be ob tained from
the Sylow xo--systems by deleting the appropriate terms.
It must be noted, however, that a group may very well satisfy C^ for
some suitable
-m
w itho ut being -nr-Sylow-regular in C unih in's sense. Fo r
example, the simple group
L
83
has a single class of conjugate subgroups
of order 84. These subgroups are dihedral, so that L
83
satisfies Ci^j. But
L
83
has two distinct classes of conjugate subgroups of order 12 and there-
fore does not satisfy C
23
. Thus
L
83
is not 2,3,7-Sylow-regular.
8/18/2019 Theorems Like Sylow's (Hall)
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THEO REM S LIK E SYLOW 'S 291
1.7. E-Theorems. Ev ery Z)-theorem includes a C-theorem and a n
jEJ-theorem by im plication. But , of course, an ^/-theorem can som etimes be
proved from hypotheses too weak to yield the corresponding D-result.
The basic ^-theorem is Schur's,
THEOREM E l . If G has a normal S
m
,-subgroup, then G satisfies E
m
.
This may be expressed in a more general form as follows:
THEOREM E l . * If K is a normal subgroup o f G such that G/K is a m-group,
then G has a m-subgroup L such that KL = G and K n L is nilpotent.
From this we may deduce the following result, suggested by Cunihin's
theorems on the factorization of -nr-separable groups (4).
THEOREM A3. Let m
0
,
in
lt
and vr
2
be mutually exclusive sets of primes.
If every c.f. of the inite group G is either a
tcr
0
m^group or else a
vr
0
m
2
-group,
then G
=
HK, where H is a -m
0
^-subgroup, K is a w
0
m
2
-subgroup,
and
H n K is a soluble m
0
-subgroup.
An immediate consequence of Theorem El is the well-known
THEOREM E 2 .
If K is a normal subgroup of G such that K satisfies G
m
and G/K satisfies E
m
, then G satisfies E
m
.
I t is no t sufficient in this theorem to assume th a t both
K
and
G/K
satisfy
E
m
. For example, the group of automorph isms L, of orde r 336, of the simple
group L = L
7
, of orde r 168, does not satisfy E
23
, although both L and L/L
do so; because the outer automorphisms of L interchange the two classes
of conjugate octahedral subgroups in L.
It will be convenient to mention here the following corollaries of
Theorem E2.
COROLLARY E2 .1 . / / all the c.f. of G satisfy C
m
, then G satisfies E
m
.
COROLLARY
E2.2. ' / /
the c.f. of G are either m-groups or else m'p-groups
for various primes p in w, then G satisfies E
m
.
COROLLARY E2 .3 . / / G is in-serial and m
x
= m, m', mq, TJJ'P, or pq,
where p is in w and q in -&•', then G satisfies E
mi
.
I t w as shown in (11) th at , if a finite grou p G satisfies E
p
. for all primes p,
then
G
is soluble. More generally, by a ve ry similar argum ent, it is easy
to see that, if G satisfies E
m
and also E
p
. for all primes p in -nr, then G is
m-separable.
It may well be conjectured that
G
is soluble whenever it
satisfies
E
pq
for all pairs of prim es
p
and
q
which divide its order. Some
light may perhaps be thrown on this question by studying the behaviour
of the better known insoluble groups in relation to the propositions E
p q
.
Here we only consider the symmetric groups and prove:
THEOREM A4. Let S
n
be the symmetric group of order
n\
and letp
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292 P. H A L L
where p and q are primes. Then S
n
satisfies E
lhq
only ivhen p = 2, q = 3 ,
and n
= 3, 4, 5, 7,
and
8.
For n = 5,1, and
8, E
u
satisfies
C|,3
but not
D
2 3
.
Taken in conjunction with Theorem D3, this result shows that, if
m
involves more than two primes not exceeding
n,
an ^-subgroup of S
?l
,
supposing one to exist, can only be insoluble. An exam ple would be the
£y-subgroups of S
p
, where p is a prime greater than 5. These are iso-
morphic with ^
p
-
x
.
1.8. C-Theorems. In a long series of papers (see particularly 2 , 3 ,5 , 6 , 7 ) ,
Cunihin proves a num ber of interesting (7-theorems, of which we m ention
two,
deducible from Zassenhaus's Theorems D6 and D7, respectively.
Let C%, be the following proposition about a finite group G.
C^: G satisfies C
m
and NJT is soluble, where T i s any ^-subgroup of
G and N is its normalizer in G.
Then the theorems in question are as follows.
T H E O R E M
C l .
If K is a normal subgroup of G such that K satisfies C
m
and G/K satisfies C
s
m
, then G satisfies C
m
.
T H E O R E M C 2. If K is a normal subgroup of G such that K satisfies C^
and G/K satisfies C
m
, then G satisfies C
m
.
I t is a corollary of Theorem Cl t hat , if G has a series whose factor groups
all satisfy C
s
m
, then G itself satisfies C
s
m
. It is a corollary of Theorem C2
that, if G has a series whose factor groups all satisfy C
s
m
, then G itself
satisfies C^. In view of Corollary E2 .1, we can also say th at , if G has a
series whose factor groups all satisfy G
m
, and if all m-subgroups of G a re
soluble, then G will satisfy
C%\
in view of Theorem El, we can add that,
if
G
has a series whose factor groups all satisfy
C
m
and if all xu'-subgroups
of
G
are soluble, then
G
will satisfy C^..
It is natural to make the
C O N J E C T U R E
C 3 .
If K is a normal subgroup of G such that both K and
G/K satisfy C
m
, then G satisfies C^.
The argument which yields Theorems Cl and C2 from Theorems D6
and D7 shows that this conjecture is equivalent to the special case already
mentioned by Zassenhaus, viz.
C O N J E C T U R E C 3 . * / / G has a normal S
m
.-subgroup K, then G satisfies C
m
.
Zassenhaus states, in (15), p. 126, th at E . W itt has shown how to deduce
C3*
from the still more special case where K is a simple group of composite
order and the centralizer of
K
in G is 1. We give here a proof of this red uc-
tion in a somewhat more precise form.
Let us say that a group G involves an abstract group V if there is a
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THEOREMS LIKE SYLOW'S 293
subgroup
H
of
G
and a normal subgroup
K
of
H
such that
H/K
~ F.
If in addition
(T) < (G), we
shall say that
G involves V properly.
We define a finite group G to be
m-exceptional
if it satisfies the following
three conditions:
(i)
G
has a normal ^/- subgroup ,
(ii)
G
does not satisfy
C
m
.
(iii) Every group F properly involved in G satisfies C
m
.
It is clear that, if
G
satisfies (i) and involves F, then F will also satisfy
(i).
Hence every group which satisfies both (i) and (ii) involves a zcr-excep-
tional group. Thus Conjectures C3 and C3* are equivalent to the state-
ment that xtr-exceptional groups do not exist. We prove
T H E O R E M A 5 .
Let
G be m-exceptional. Then both the normal S
m
.-subgroup
K of G and also G/K are simple groups of composite order and the centralizer
of
K in G is I, so
that
G may
be regarded
as a
subgroup
of
the group
of
auto-
morphisms of K. For any given prime divisor q of (K), there exist S
m
-sub-
groups
of G
which
do not
leave invariant
any
Sylow q-subgroup
of K. G
satis-
fies C
mg
but not D
mq
.
A closer analysis of the properties which a hypothetical -nr-exceptional
group
G
must have, particularly those resulting from (iii) above, would no
doubt be of some interest. Here we merely stat e:
THEOREM C4.
Let K be normal in G and suppose that both K and G/K
satisfy C
m
. Let T be an S
m
-subgroup of K, M its normalizer in G and
N = KnM. Then M/N ~ G/K. If S/N is an S
m
-subgroup of M/N, then
either G satisfies C
m
or else S/T involves a m-exceptional group.
In view of Theorem A5, this result contains both Theorems Cl and C2
as special cases. I t is a corollary of Theorem C4 th at, if
G
has a series
whose factor groups all satisfy
C
m
,
then either
G
satisfies
C
m
or else
G
involves a -or-exceptional group.
As Zassenhaus remarks (loc. cit.) the Conjectures C3 and C3* would
follow from either of the long-standing conjectures:
(a)
All groups of odd order are soluble.
(6) The group of outer automorphisms of a finite simple group is soluble.
They would also follow, as is shown by Theorem A5, from the conjecturef
(c) If A is a subgroup of the group of automorphisms of the finite group G
and if
(A)
and
(G )
are coprime, then for every prime
q
dividing
(G),
A
leaves
invariant some Sylow ^-subgroup of
G.
That (c) is true if either
A
or G is soluble follows easily from Zassenhaus's
f That the conjecture C3* is equivalent to (c) is proved by D. G. Higman, Pacific
J.
Math. 4 (1954), 545-55. I am indebted to the referee for this reference.
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294 P . HALL
theorems
D6 and
D 7.
I n
this connexion
it may be
of some intere st
to
point
ou t
th e
following consequence
of
Theorem
D 5.
T H E O R E M
A6.
If 0 is a finite group satisfying E^ and A is a soluble
m-subgroup of the group of automorphisms of G, then A leaves invariant some
S
m
-subgroup of G.
For we can take G in its regular representation and regard AG as a
subgroup of the holomorph of G. By Theorem D5, A G satisfies D
m>
since
AG/G r^j A and so satisfies D^, while G satisfies
E%.
Hence A is contained
in some #
OT
-subgroup HoiAG. Therefore A leaves invariant the ^- su bg ro up
H n G of G.
We remark tha t, if A is a soluble subg roup of the group of autom orphism s
of the arbitrary finite group G and if there is at most one prime q which
divides both (A ) and (G), then we can take vr in Theorem A6 to consist of
the prime divisors of (^4) together with
q
if necessary. In this case then ,
A
leaves invariant some Sylow ^-subgroup of
G,
no special assumption
about
G
being required.
W e hav e already n oted (Corollary E2.3) th at m-serial groups satisfy
E
m
.
I t w ould follow from Conjecture C3* th a t they even satisfy D
w
. In fact
we have:
T H E O R E M D 8. Let G be a nr-serial
group.
If L is a soluble m-subgroup
of G and if H is an S
m
-subgroup of G, then L
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THEO REM S LIK E SYLOW 'S 295
We may therefore suppose that K is nilpotent and have only to prove
the existence of a -nr-subgroup S such that KS = G. If there is a normal
subgroup
K
x
of G such tha t 1 <
K
x
<
K,
we apply the induction h ypothesis
to G/K
x
and obtain a subgroup S
x
^ K
x
such that S
x
/K
x
is a tr-group and
KS
X
= G. Then we apply the induction hypothesis to S
x
and obtain a
-nr-subgroup S such that K
x
S = S
x
. This gives KS = KK
X
S = KS
X
= G
as required.
Finally, let K he a, minimal normal subgroup of G. Being nilpotent, it
is an Abelian #>-group. If
p
is in -nr, we can take
S
=
G.
If
p
is in -a/,
then K is a norm al Abelian /S^-subgroup of G. B y Theorem D2, G satisfies
E
m
and we can take for
S
any /S^-subgroup of
G.
Proof of
A3. We begin with an obvious rem ark. Let
G=G
0
>G
x
>...>G
r
=l
be any series of subgroups of G. Let H and K be subgroups of G such
Then
hat , for each
i
= 1,2
G = HK.
For, by hypothesis,
,...,
r,
e i ther G
e i the r G
r
_
x
^
r
t
_
x
0, that
G
t
^ J?iT. If G ^ ^ ^JEf,
then also G^- ̂ < ^ ^ , since ^ is normal in
G^^
Hence
O
t
_
x
G
x
> ... >
G
r
=
1 be a chief series of
G,
so that each
G
€
is norm al in
G.
Ev ery factor group G^JGi is a direct produ ct of isomorphic
c.f. of
G.
Hence we may divide these
r
factor groups into two mutually
exclusive classes, putting in the first class those which are TO-
0
xo-j-groups
(including those, if any, which are xn-
0
-groups) and in the second class all
the rest. We now show th at th ere is a m
0
to^-subgroup H and a TO-
0
'OT
2
-
subgroup
K
in G such tha t (i) if
G
i
_
x
lG
i
is in the first class, the n G
{
_
x
^ G
i
H,
while K^JKi is nilp ote nt, and (ii) if G
i
_
x
IG
i
is in th e second class, th en
G^
x
^
G
t
K
and
H^JHt
is nilpotent. Here we have written
H
i
= G
i
C\ H
and
K
i
=
6^ n
K.
Arguing by induction on r, we m ay assume the ex istence
of subgroups
H
and
K
containing
G
r
_
x
and such that
H/G
r
_
x
is a vr
0
m
x
-
group, K/G
r
_
x
is a
vr
0
nr
2
-group and (i)* if G^JGj is in the first class and
i
<
r,
then
G
t
_
x
<
G^
while
K
i
_
x
IK
i
is nilp ote nt, (ii)* if G^JGi is in the
second class and i
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296 P. HALL
now
G
r
_
1
/G
r
~ G
r
_
x
is in the first class, we take H = H, while for K we
take a vr
Q
-nrg-subgroup of K such that
G
r
_
x
K
= K and G
r
_
x
n K is nil-
potent. Such a subgroup K exists by Theorem El*. Then it is clear from
(i)*
and (ii)* that
H
and
K
satisfy (i) and (ii) for each
i =
1,
2,..., r.
Also
H
=
H
is a xtr
0
TDygroup.
Similarly if
G
r
_
x
/G
r
is in the second class.
The remark made above shows that
G = HK.
The intersection
is a TD-
0
-group since vr
Q
, m
v
and vr
2
are mutual ly exclusive. If D
i
= G
t
C\ D,
then
D
i
_
1
ID
i
~ G
i
Di-JG^ which is a subgroup of both G
i
H
i
_
1
IG
i
~ H^JH^
and of G
i
K^JGi ~ K^JK^ But for each i, one of the two groups H^JHi and
Ki.JKj is nilpotent. Hence all the factor groups D
i
_
1
/D
i
are nilpotent
and D is soluble. Thus A3 is completely proved.
2 . 3 . LEMMA 1.
Let K be a normal subgroup of G and let H be an S
m
-
subgroup of G. Then K
n
H is an S
m
-subgroup of K and KH/K is an
S
m
-subgroup of G/K.
For let (G:K) = h and (K) = k. Then
{G)
m
= h
m
k
m
= (H) = (H:Kn H)(K
n
H).
But
H/K
n
H ~ KH/K,
which is a w-subgroup of
G/K,
so that
(H:Kn H)
divides h
m
\ and K n H is a To--subgroup of K so that (K n H) divides k
m
.
Hence (KH:K) = h
m
and (K n H) — k
m
, as required.
Proof of E2. Let K satisfy C
m
and let 6?/Z satisfy E
m
. Let T be an
/S^-subgroup of
K
and let
N
be its normalizer in
G.
For any
x e G, T
x
is
an /S^-subgroup of iT and hence is conjugate to T in K so that T
x
= T
v
with y e . Thus
xy-
1
e N
and
NK = G.
Hence JV/Z n
N
~
G/JK" SO
that
JV/X n JV has an #
OT
-subgroup H/K n JV. Apply Theorem El to the group
H/T with the normal ̂ --subgroup K n
iV/2
7
and we obtain an ^-subgroup
U/T of H/T. Then (U:T) = {H.K D N) = h
m
and (T) = /c
OT5
so that U
is an /S^-subgroup of G.
L E M M A
2. .Le£
H = H
x
xH
2
x
...xH
r
.
If each H
t
has a given one of the
properties E
w
, C
w
, D
m
, E*
m
, C
s
m
, D
s
m
, E , or C^, then H has the same property.
The proof is straightforward and may be omit ted .
Suppose now that all the c.f. of G satisfy C
m
, and let K be a minimal
normal subgroup of G. Then K is a direct product of c.f. of G. Hence K
satisfies C
m
by Lemma 2. Corollary E2.1 now follows from Theorem E2
by an induction argument on (
0),
since we may assume tha t
G/K
satisfies
E
m
.
Since trr-groups satisfy C
m
trivially and
TO-'^-groups
satisfy C
m
by Sylow's
theorem, we obtain Corollary E2.2 as a particular case of E2.1.
As for Corollary E2.3, this follows from E2.2 for
m
1
= w, m', vrq,
and
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298 P . HALL
We make a few pre lim inar y simplifications. Since G/K satisfies D
m
, we
have L
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THEOREMS LIKE SYLOW'S 299
Now let
Q
be the normalizer of
P
in
G
and let
Q
=
K
n
Q.
We first
prove (5) that Q satisfies El^. Since P and Q are both normal in Q and
Pn Q ^. Lr\ K = I,
we have
PQ = PxQ.
Let the distinct primes
dividing (G)^ be p = p
x
, p
2
,..., p
r
and let Q
i
be a Sylow
r
subgroup of Q
for
i —
1, 2,..., r. Since T satisfies Z^ we may choose an iS^-subgroup T$
of T such tha t P x (^ < T,. We note that ^ = Z n 2J is an ^ -s ub gro up
of̂ fiT by Lemma land
is
therefore nilpotent. Also
T
= i£ P and
so 7^
=
S
i
P.
Hence T
t
contains a unique Sylow ^-subgroup P^ containing P and the
centralizer
Z
t
of
P
t
in
T
t
is nilpotent and contains
Q^
Since
T
satisfies
D
mi
the subgroups T̂ are all isomorphic. More precisely, there is an isomorphism
mapping
T
t
on to
T
x
and at the same time mapping
P
t
on to
P
x
and therefore
Zi onto Z
x
. If this isomorphism maps Qi onto ii^for i = 2,
3,...,
r, itfollows
that iẐ <
Z
x
.
Since
Z
x
is nilpotent, the product
R
of
R
x
= Q
x
,
R
2
,...,
R
r
is direct; and since P -group,
P
< P
x
would imply that
P < $ n
P
lt
so that
P = LOT
would not be an ^- su bg ro up of
Q
n T
7
and therefore, by Lemma 1, L would not be an iS^-subgroup of Q. Hence
P =
P
x
is a Sylow ^-subgroup of
H
n T. Also
H 0 T is
normal in J?. By
Sylow's theorem, the normalizer Q C\ H = U of P in H satisfies H ^. TU.
Since T =
KP
and P <
U,
we even have
H ^KU.
Since
KH = G by
(2), we have i£?7 = £. But LnK = 1. Hence (17) > (X). But £ is an
/S^-subgroup of
Q
and
?7
is a -or-subgroup of
Q.
Hence
U
is an /S^-subgroup
of Q. But Q satisfies D
m
by induction. Hence L = U
x
^ ^
x
for some
x e G
and Theorem D5 is completely proved.
Corollary D5.1 follows immediately. If G has a series all of whose factor
groups satisfy
E^,
then all the c.f. of
G
satisfy E
1
^ by Lemma 1. If
K
is
a minimal normal subgroup of G, then K satisfies E
1
^. by Lemma 2. The
c.f. of
G/K
all satisfy
E%
and we may suppose inductively that
GjK
satisfies
D
s
m
.
Then
G
satisfies
D%.
by Theorem D5.
Corollary D5.2 is a particular case of D5.1. For if
G
is -nr-separable and
vr
1
is any non-empty subset of
w,
then every c.f. of
G
is a
w'
x
p
-group for
some prime
p
in
-m
x
and therefore satisfies
E^
by Sylow's theorem.
Corollary D5.3 is a special case of D5.2.
Proof of A1. Suppose H and K are S
m
-subgroups of G and that both H
and
K
have Sylow series of complexion
(p
1
,P2>-'->P
r
)>
where
p
x
,..., p
r
are the
distinct primes dividing (G)
m
. If H
x
and K
x
are the normal
S
VlPi
Prl
-
subgroups of
H
and
K,
respectively, then
H
x
and
K
x
have Sylow series of
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300 P. HA LL
complexion (Pi,.--,Pr-i)
a n
d
a r e
$p
lf
2>,
3
»
r
_
1
-subgroups of
G.
Arguing by
induction on r, we may suppose
H
x
and
K
x
are conjugate in
G.
We may
then take
H
x
= K
x
,
replacing
K
by a conjugate if necessary.
H/H
x
smd.K/H
x
are then two Sylow ̂ -su bg rou ps of
N /H
x
,
where
N
is the normalizer of
H
x
in
G.
Hence
H
is conjugate to
K
in
N.
2.5. Proof of
D7*. I t will be sufficient t o show th a t a ttr'-soluble grou p
G
satisfies
D
m
.
Let
K
be a minimal normal subgroup of G. Then
K
is either a -nr-group
or else an Abelian g-group for some prime
q
in
m'.
Also
G/K
is icr'-soluble.
Arguing by induction on
(G),
we may assume th at
G/K
satisfies
D
m
.
If
K
is a -nx-group, then
G
satisfies
D
m
by Lemm a 4. Thus we may assume t ha t
K
is an Abelian -nr'-group.
G
satisfies
E
w
by Theorem E2. Let
H
be an
^ - s u b g r o u p o f
G
and
L
any xtr-subgroup of
G.
Since
G/K
satisfies
D
m
by
the induction hypothesis, we have L
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THEOREMS LIKE SYLOW'S 301
K cannot be Abelian or G would satisfy C
m
by Theorem D2. Hence
K
=
K
1
xK
2
X...xK
r
,
where the
K
€
are isomorphic simple groups of com-
posite order. The
r
subgroups iQ are the only normal subgroups of
K
which are simple. On transforming K by the elements of H, the K
t
must
therefore be permuted transitively among themselves. Hence if N is the
normalizer of
K
x
in
G,
we have
K
<
N
and
(G:N) = r.
Let
q
be a prime
divisor of (Kj) and let Q
x
be any Sylow (/-subgroup of K
x
. Let N be the
normalizer of Q
x
in N. By Sylow's theorem, K
X
N = N and therefore
KN = N. Hence N/K n N ~ F / Z and, by Theorem El , 2V contains an
£
OT
-subgroup
U,
so that
{K n N)U = N
and therefore
KU = N.
Thus £7
is an iS
m
-subgroup of N. Since 6r = iT£T = KH
X
and K ^. N, the inter-
sections V = H r\ N and 1̂ = T̂
x
n iV are also ^. -subgroups of N.
Suppose if possible that
r
> 1. Then
N < G
and hence
N
satisfies
C
m
.
Consequently
U, V, V
x
are conjugate in
N.
Replacing / / and
H
x
by con-
jugates, we may assume tha t U = V = V
x
. We then have (H:U) = r and
may choose the elements h
x
= 1, h
2
,..., h
r
in H so that
. £*
= jfî (i = i, 2,..., r).
Then Qi* = Qi is a Sylow ^-subgroup of
iQ
and hence
Q = Q
1
xQ
2
X...xQ
r
is a Sylow ^-subgroup of K. For any xe H, we have ^x = u^-, where
w
e
C7
and j = j (i ,3 ). Hence Qf =
Q^
x
= Q * = Q$ = Q
p
since
U
<
N.
For a fixed xe H, i -+j(i,x) is a permutation of 1, 2,..., r. Hence Q
x
= Q
and so i/ is contained in the normalizer R of Q in 6r. Similarly H
x
is con-
tained in R. Since $ is not normal in K, we have R < G and so 72 satisfies
C
m
. Hence the ^-s ubgro ups H and H
x
of i? are conjugate, a contradiction.
Thus r = 1 and i£ is a simple -nr'-group of composite order.
By Sylow's theorem
KR = G,
so tha t the ^-subgroups of
R,
which are
all conjugate in R as we have already remarked, are also ^- su bgr ou ps
of
G.
There is therefore a uniquely determined class of conjugate
^-sub group s of G associated with any given prime divisor q of (K), viz. those
which belong to the normalizer in
G
of some Sylow (/-subgroup of
K
. Since
G does not satisfy C
m
, there must be some S
m
-subgroups of G which do
not leave invariant any Sylow (/-subgroup of
K.
If
H
is one of these, then
H cannot be contained in any
m(2
-subgroup L of G, for L n K is normal
in
L
and is a Sylow (/-subgroup of
K.
Thus
G
cannot satisfy
D
mq
.
But
G satisfies E
mq
bj Corollary D5.3, or more simply by Sylow's theorem and
Theorem El. In fact, G satisfies C
OT(r
For let L and L
x
be any two
/S^-subgroups of G. Then LP\K and L
X
D K are Sylow (/-subgroups of K and
therefore conjugate. Replacing L
x
by a conjugate if necessary, we may
suppose that Lc\ K = L
x
n K = Q, so that both L and L
x
belong to the
normalizer R of Q. L/Q and LJQ are then ^-s ubgrou ps of R/Q and
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302 P. H A L L
therefore conjugate because R/Q is properly involved in G and so satisfies
C
m
.
Hence
L
and
L
x
are conjugate.
Let Z be the centralizer of K in
G.
Then Z is norm al in 0 and K n Z = 1
because if is simple bu t not cyclic. Since G/K is also simple, Z ^
1
would
imply that G = KxZ. This would make Z a normal and therefore the
unique ^-subgroup of G. This is impossible since G does not satisfy C
w
.
Hence
Z
= 1 and Theorem A5 is completely proved .
Proof of C4. Let K be a normal subgroup of G such that both K and
#/ £ satisfy C^. By Theorem E2,
G
satisfies
E
m
.
Suppose th at G does not
satisfy
C
m
and let
H
and
H
x
be
/S .̂
-subgroups of
G
which are not con-
jugate in G. The ^ - subgroups H n K and H
X
C\ K of K are conjugate and
we may assume H f) K = H
X
C\ K = T, replacing H
x
by a conjugate if
necessary. Then both
H
and
H
x
are contained in the normalizer
M
of
T
in G. Let N = K n M. Since Z satisfies C
OT
, we have KM = G as in the
proof of Theorem E2 . Hence M/N ~ £ / Z and satisfies C
w
: The
^ - subg roups NH/N and NHJN of Jf/iV are therefore conjuga te and we
may assume that
NH = NH
X
=
S,
replacing
H
x
by a conjugate if neces-
sary. S/T has the normal ̂ - su b gr ou p N/T and has the two non -conjugate
S^-subgroups
H/T
and
HJT.
Hence
S/T
does no t satisfy
C
m
.
Therefore
S/T
involves a w-exceptional group and Theorem C4 is prove d.
2.7. Proof of D8. We argue by induction on (G). Let K be a minimal
normal subgroup of
G, H
any /8^-subgroup of
G,
and
L
any xo--subgroup
of G. We have to prove that, if either L is soluble or if G involves no
•nr-exceptional group, then
L
^
H
x
for some
x e G.
Since
KL/K
is a
-nr-subgroup and KH/K is an /S^-subgroup of G/K, we may suppose by
induction that
L
<
KH
X
for some
x
e
G;
for if
L
is soluble, so is
KL/K,
while if (3 involves no xir-exceptional group, neither does G/K. Also, KH
X
is a -nr-serial group and involves no -nr-exceptional group unless G involves
one.
The result now follows by induction if KH
X
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303
2.8. Proof
of A4. Let p < q
<
w, where p
and
q
are
primes,
and let
H
be an $
pg
-subgroup of the sym metric group S
n
. By a theorem of Burn side,
H is
soluble.
Suppose first that H is a primitive permutation group. Then n = r
m
,
where r is
a
prime, an d r is either p
or q.
Also, H is contained
in
the holo-
morph H of the elementary Abelian group of order r
m
. Cf. Speiser (13,
Satz 98). Since
(H)
r
= r^-
m(
-
m+x
>
and
(H)
r
= ( SJ
r
=
r
i+r+r*+...+r»-»
} w e m u s t
h a v e
£
This equation implies that either m =
1
or else
m =
2,
r =
2. In the case
m
=
1,
if is the
metacyclic group
of
order r(r—1),
so
tha t
we
must have
r =
q = n =
l-\-kp. But then (?>
n
)
p
^
p
k
,
so t ha t
&
must be divisible by
p*-
1
.
Thus either k =
2,
p =
2, or
k.=
1. But
(S
5
)
2
= 8 so
tha t &
=
2
is
impossible. Hence jfc=l,^>
= 2, g = 3, and
H =
S
3
.
If, on the other hand, m = 2, so th at r = 2 = p, we have w = 4 and so
q = 3, H = S
4
. We conclude tha t the only cases where H can be primitive
are when p = 2, q = 3, and w = 3 or 4, and in these cases H is the whole
of S,
/r
In all other cases
H
is either imprimitive or intransitive.
Ne xt suppose th at H is intran sitive , with com ponents of degrees
I
and m
where £+ra
=
n.
Then
i/ is
contained
in S
z
x
S.
m
and, since
it
must be
an
$p
S
-subgroup
of
this group, both
£j and S
m
must satisfy
JE^.
Further,
# cannot exceed both I and m, since the n (H ) would be prime to q, contrary
I
of
S; X S
wl
in S
M
must
be
prime
to
pq,
mj
since otherwise H could
not be an
$p
g
-subgroup
of
S
M
.
A similar argument applies
if
JÊ
is
imprimitive, with
e
systems
of
imprimitivity of d symbols each. Then n = de and H is contained in
S^ISg,
a
maximal imprimitive subgroup
of S
M
corresponding
to
this
factorization of n. The order of 2^1; £
e
is (d\)
e
e\ and therefore q cannot
exceed both d
and e.
Since G
=
S
d
1 S
e
has a
normal subgroup K which
is the direct prod uct of e factors isomorphic with 2
d
and G/K ~ S
e>
it is
clear from Lem ma 1 th at bo th
2
d
and 2
e
must satisfy E
pq
. Further
the
index n\/(d\)
e
e\ of G in S
n
must be prime to pq.
If therefore JET
is
either imprimitive
or
intransitive, there
is
necessarily
an integer n
x
such that q
8/18/2019 Theorems Like Sylow's (Hall)
19/19
304 T H E O R E M S L I K E S Y L O W ' S
A subgroup
of
index
35 in S
7
c a n n o t
be
t r ans i t ive
and so
m u s t h a v e
the form E
3
x S
4
. Again
we
h a v e
a
single class
of
#
2 3
- subgroups . None
of
these con ta ins
a
cyc l ic pe rmuta t ion
of
order
6.
T h u s
S
7
satisfies
C
23
but
n o t D
2
3
.
A subgro up of index 35
in S
8
c a n n o t
be
in t ran s i t ive s ince
all th e
b inomia l
coefficients 8, 28, 56 ,70 ar e even .
I t
cann ot be pr im it ive s ince th e ho lom orph
of
the
e le m e n ta r y g r o u p
of
order
8 has
index
30 in S
8
.
H e n c e
it
m u s t
be
im p r im i t i v e
and
on ly
the
subgroups
of the
form S
4
1 S
2
are
la rge e noug h.
Aga in
we
h a v e
a
single class
of
conjugate
S
2
3
-subgi\oups. None
of
t h e m
con ta ins
a
s u b g r o u p
of
th e form 2
2
1 E
4
. Thus
2
8
satisfies C
2 3
bu t not
Z )
23
.
I t r ema ins
to
show tha t ,
if n > 8, E
n
does
not
satisfy
E
2 3
. Suppose
the
c o n t r a r y
and
choose
n > 8 as
smal l
as
possible
so
t h a t
2
n
has an
S
2 3
- subgroup H. If H is in t rans i t ive w i th componen ts of degrees I and m,
w h e r e I
d
%
S
e
where
de
= n. As we have
also seen, both 2,
d
and 2
e
must have $
23
-subgroups and n\/(d\)
e
e\ must be
prime to 6. By our choice of n, both d and e must be ^ 8, ^ 6, and ̂ 1.
But in this case H is transitive so that n must be of the form 2
a
3^. Thus
d
and e can only be 2, 3, 4, or 8. Again it is easy to verify that none of the
eleven relevant indices
is
prime to
6.
This concludes the proof of Theorem
A4.
REFERENCES
1.
W. BURNSIDE, Theory
of
groups
of
finite order, 2nd edition (Cambridge, 1911).
2.
S.
A.
CUNIHIN,
Mat. Sbornik,
N .S . (25) 67
(1949), 321-46.
3.
Doklady
Akad.
Nauk S.S.S.R.
N.S.
73
(1950), 29-32.
4. ibid.
95
(1954), 725-7.
5. ibid.
66
(1949), 165-8.
6. ibid.
69
(1949), 735-7.
7. Mat. Sbornik, N.S.
(33) 75
(1953), 111-32.
8.
P.
A.
GOL'BERG, Doklady
Akad. Nauk S.S.S.R.
N .S . 64
(1949), 615-8 .
9.
P.
HAXL,
J. London Math. Soc. 3
(1928), 98-105.
10. Proc. London Math. Soc. (2) 43
(1937), 316 -23.
11. J . London Math. Soc. 12
(1937), 198-200.
12. and GRAHAM HIGMAN, Proc. London Math. Soc. (3) 6
(1956),
1-42.
13.
A.
SPEISER,
Theorie der Gruppen
von
endlicher Ordnung,
3rd
edition (Berlin,
1937).
14.
H. WIELANDT,
Math. Zeitschrift,
60
(1954), 40 7-8 .
15.
H. ZASSENHAUS,
Lehrbuch der Gruppentheorie (Leipzig and Berlin, 1937).
King's College,
Cambridge