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FGRMATEONS AND RELAEVE
£3 NURMALIZERS

Thesis for the Begree of Ph. D.
MiCHmN STATE UNWERSITY
JGHN DAVE GILL‘W
1970

 

fi‘J

LIBRARY ‘

Michigan State
University '

  
    

«"4ng

This is to certify that the
thesis entitled

Formations and Relative 8 normalizers

presented by

John David Gillam

has been accepted towards fulfillment
of the requirements for

 

Ph .D . degree in Ma ghematics
:fi. W M
Lee M. Sonneborn
Major professor
l- 13 - 70

Date

 

0-169

  
   
     

 
  
 

BINDING DY
, ms & sons
‘j max swam me. ,

LEM!!! EEEEDSRS

ABSTRACT

FORMATIONS AND RELATIVE
3 NORMALIZERS

BY

John David Gillam

In this thesis, G denotes a finite solvable group; 3 is
a formation locally defined by {3(p)} with 3(p) ; 3 for all
primes p, and "(3) = {p: 3(p) # ¢}.

Definition. Let was. A p-chief factor H/K of G
is N-S central if N/CN(H/K) 6 3(p), and is N-S eccentric
otherwise.

Definition. Let Sp be a Sylow p-complement of G, N'd G,
and CpflN) the intersection of the centralizers of the 3 central
p-chief factors of N. Let X(SP,N) = NG(SP 0 0pm) if 3(9) i‘ q).

and X(SP,N) = Sp

if 3(p) = ¢. Let 2 be a set of Sylow p-
complements of G. A relative 3 normalizer (with respect to N)
is the subgroup DNGJ) = n{X(Sp,N): Sp 6 2}. Also let
TN(2) = fl{X(Sp,N): Sp 6 E and p E n(3)}-

The following statements are the main re8ults obtained in
this thesis.

Theorem. DN(2) covers N-3 central chief factors of G
and avoids N-3 eccentric chief factors of G. Th(2) avoids

N-3 eccentric n(3)-chief factors of G and covers all other

chief factors of G.

John David Gillam

Theorem. If e is an epimorphism of G, then
(Du (2))9 = DNeme) and (TN(2))9 = 13,9029).

Theorem. Let N 4 G. Then N E 3 if and only if N is
a n(3) group and all n(3)-chief factors of G are N-3 central.
Hence if z is a set of Sylow p-complements of G, N E 3 if and
only if N is a 11(3) group and TING) '3 G.

Theorem. Let N be a normal n(3) subgroup of G. If
N3 is complemented in N by an 3 projector of N, then N3
is complemented in G. Furthermore all complements of N3 in
G are conjugate and are precisely the various TN(2).

Example. Let G be a solvable Frobenius group with
Frobenius Kernel K. Then K is complemented in any solvable
extension of G, and all such complements are conjugate.

Theorem. Let N 4 G; than N E 3 if and only if N is
a 11(3) group and N§(G)/§(G) 6 3-

Corollary. Let D, M14 G with M a n(3) group and
D s: @(c). If M/D E 3, then M e 3.

Theorem. Let N 4 G. Then N E 3 if and only if N is
a n(3) group and all n(3)-chief factors of G between §(G)
and Fit(G) are N-3 central.

Theorem. Let N <1 G. Then. N e 3 if and only if N is
a 11(3) group and for p 6 11(3) with 3(p) c: 3
M n N/Core (NOW 6 3(p) for all p-maximal subgroups M of G.

Example. Let N 4 'G and 11 a set of primes. Then N
is p-nilpotent for all p E n if and only if M11 N C Core(M)

for all n-maximal subgroups M of G.

John David Gillam

Theorem. Let 2 be a set of Sylow p-complements of G.

Then D B Dam) is an 3 projector of N603 n G3).

Corollary. Let 2 be a set of Sylow p-complements of G.
Then DG(Z) is an 3 projector and an 3 normalizer of

NGQ‘. 0 G3)“ If G o! 3. then Dc(z:) c NG(2 n c and

)
8
G =‘<NG(2 fl G3)g: g E G>e Hence every finite solvable group is
generated by subgroups in which 3 projectors coincide with 3

normalizers.

FORMATIONS AND RELATIVE
3 NORMALIZERS

BY

John David Gillam

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1970

(962273!
7~/~7o

To Mary Lee, my wife.

ii

ACKNOWIEDGEMENTS

I should like to express my appreciation to Professor
Lee M. Sonneborn, without whose advice, encouragement, and
friendship this thesis would not have been written. Also, I

am grateful to Professor Ti Yen for his many helpful suggestions.

iii

TABLE OF CONTENTS

CHAHER I O O ...... O O O O O ..... O O O O O O O O O O O O O O O O O 00000 O O ......... 1
CMPI‘ER II I O O O O O O O O I I O O O O O I O O O O O O O O O O O O O O O O O O O O O O O O 0 ...... O I 4
BIB LImRAHiY O O O O O O O O O O O O O O O O I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 2 2

APPWDIX OOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOO00.00.000.000.0. 23

iv

Chapter I

All groups in this thesis are finite and solvable. Unless
otherwise indicated, G will denote an arbitrary finite solvable
group. A chief series of G is a G-composition series. A chief
factor of G is a composition factor of some chief series of G.

If n is a set of primes, a n-chief factor of G is a chief factor
whose order is a power of some prime belonging to n; a n-maximal
subgroup of G is a maximal subgroup whose index in G is a

power of some prime belonging to n. If H/K is a chief factor

of G and M is a subgroup of G, then

CM(%) I {m E‘M; th = hK for all h E H}; M complements E' if

K
G c‘HM and H n M c K. A subgroup u| of G covers a Subgroup
g of 5 if A : MB (equivalently, A .. (M n A”); M avoids 3.5

if M n A c B. A subgroup M of G is a cover-avoid subgroup
if M covers or avoids every chief factor of G. Let
2: = {Sp: p ‘ o(G)} be a set of Sylow p-complements of G. If M
is a subgroup of G, 2 is reducible to M if Sp 0 M is a Sylow
p-complement of M for all Sp 6 2. Throughout, the intersection
of a vacuous collection of subgroups of G will be interpreted
as G.

A formation 3 is a class of groups such that:
1). If G E 3, then every epimorphic image of G is in 3.

2). If cm1 and GIN2 are in 3, then G/Nlnnzes.

A non-empty formation 3 is saturated if G/§(G) in 3 implies
G E 3. If 3 ¢ ¢, the 3 residual of G is the unique minimal

normal subgroup G such that G/G3 E 3. Note that if e is an

3

epimorphism of G, (GS)6 = (G6)3. A subgroup F of a group G
is an 3 projector of G if F E 3, and whenever F C U c G,
U = USP. Let 3(p) be a formation for each prime p. We define

3 by the following conditions: G E 3 if:
1). If 3(p) = ,3, p I 0(6).
2). If 3(p) # ¢ and H/K is a p-chief factor of G, then
G/CG(H/K) E 3(p).
If 3 is defined as above, we say 3 is locally defined by
{3(p)}. Gaschfitz and Lubeseder have shown [5; 7.5, 7.25 pp. 697,
715] that a formation 3 of solvable groups is saturated if and
only if 3 can be locally defined. Carter and Hawkes [1, p. 177]
have shown that every saturated formation 3 can be locally
defined by {3(p)} with 3(p) : 3 for all primes p. Throughout,
3 will denote a formation locally defined by 3(p) with 3(p) c 3.
We denote by n(3) the set of primes {pz 3(p) # ¢}.

The following three propositions are part of the folklore
of finite solvable groups; however, the exact statements and
proofs do not seem to appear in the literature.

Proposition 1.1. Let H/K be a chief factor of G and
M.c G such that M. covers H/K and G/GG(H/K). Then HnM/KnM
is a chief factor of M, CMGIOM/mM) - M n 66%), and
c/cG (H/K) ‘1 M/CMOI’tM/KOM).

Proof. Since H = (HnM)K, anulmnu #‘<l>. Suppose

KnM : LC mM, where L is M invariant. Since G = CG(%)M,

LK <1 G. If LK = H, then L = L(ml() 3 MILK = mu. Therefore
LK = K and L = KnM. Clearly M n ch/K) c cMalnM/KnM), and
since H = (HfiM)K, we have equality. The last assertion is now
clear.

Proposition 1.2. Suppose M is a cover-avoid subgroup
of G and 9 is an epimorphism of G. Then M9 is a cover-
avoid subgroup of GO. If M is such that M covers a chief
factor H/K of G if and only if M covers every chief factor
G-isomorphic to H/K, and H/K is a chief factor of G with
He 1‘ K9, then M9 covers He/Ke if and only if M covers H/K.

Proof. Let N = Ker(e) and H/N/K/N a chief factor of
G/N. If M avoids H/K, then MN/N n H/N = Mann/N = (mum/N
C K/N. If M covers %, H c KM C K(MN), and so m/N covers
H/N/K/N. Assume the additional hypothesis on M, and let H/K
be a chief factor of G with HN # KN. Then HN/KN is G-
isomorphic to H/K. Hence M covers H/K if and only if M
covers HN/KN, if and only if MUN covers HN/N/KN/N.

Proposition 1.3. Let 6 be an epimorphism of G, N a
normal subgroup of G, and H/K a chief factor of G such that
H6 1‘ K9. Then (cNm/xne = (heme/K9” and N/cN (H/K) ‘3 N9/CN6(He/Ke).

Proof. Let M = Ker(e). Since HM 3‘ KM, M does not cover
H/K. Hence (I-hM)K is a normal subgroup of G contained properly
in H. Therefore (mM)K = K. Since [H,M] C H n M, M C CGCH/K).
Clearly cum/loam c CHM/”(HM/M/KM/M). Let c be the pre-image
of enema/1(9). Then [C,H] 1: KM n H = Kenn) = K, and so
c s: cNMaI/K) .. CG(H/K) 0 NM = CNCH/K)M. Hence (CN(H/K))9
- emote/K9). Also N/CNCH/K) a N/(NnM)CN(H/K) ‘I NM/CN(H/K)M
1' Ne/(cN (H/K))e = N6/CN6019/K6).

Chapter 11

Definition 2.1. Let N a c. A p-chief factor H/K of
G is N-3 central if N/CN(H/K) E 3(p), and is N-3 eccentric
otherwise. If N -‘G, we say 3 central and 3 eccentric,
respectively.

Carter and Hawkes [1, 2.2, p. 178] have shown that if 3
is locally defined by {31(p)] and also by {32(p)}, then
031(9) “932(9) for 811 primes p- (031(1)) - {6: G/OP(G) e 31(9)}.
1 = 1,2.) If N 4 G and H/K is a chief factor of G, then
H/K is G-isomorphic to a chief factor H1/K1 of G with N c K

1

or H1

N-3 central. Hence to see that Definition 2.1 is independent of

C N, and H/K is N—3 central if and only if H1/K1 is

the local formations defining 3, we need consider only the chief

factors of G above and below N. Let 3 be locally defined by

{31(p)} and by {32(p)}. Then 31(p) = ¢ if and only if 32(p) = ¢.

Let 31(p) f ¢ and H/K a p-chief factor of G above or below

N. If N c K, then N/CNOI/K) =<1> s 31(1)) n 32(p). If H c: N,

then Proposition 2.2 below and the well-known fact [5, 5.17, p.

485] that Op(G) "<L> for any group G possessing a faithful

irreducible representation over a field of characteristic p

imply that N/CN(H/K) E 31(p) if and only if N/CN(H/K) 6 32(9)-
Proposition 2.2. Let N<x c and H/K a p-chief factor

of G with H C N. Then:

i). H/K is N-3 central if and only if all chief factors of
N between H and K are 3 central (in N).
ii). Chief factors of N between H and K are all 3 central
or all 3 eccentric.
iii). cam/K) - O{CN(L/M): L/M is a chief factor of N between
H and K}.
Proof. 1). Suppose H/K is N-3 central. Since
CN(H/K) G CN(L/M) for any chief factor L/M. of N. between H
and K, N/CN(L/M) is an epimorphic image of N/CN(H/K). Hence
L/M is 3 central. Conversely, suppose all N chief factors
between H and K are 3 central. By Clifford's Theorem
[5, 17.3, p. 565], H/K '6) E H

1
(1)
factors of N. Then CN(H/K9 ' n Ch(H1/K), and so

(i)

/K where the Hi/K are chief

N/cN (H/K) e 3(p) .

ii) and iii). As in i), let H/K -o 2 H

l
(1)
all conjugate in G, and hence one HilK is 3 central if and

/K. The Eli/K are

only if all Hi/K are 3 central. Also CN(H/K) ‘ 3]) qN(H1/K).
The Jordan-Hdlder Theorem applied to the N module H/K now
yields both ii) and iii).

The following definition is due to Carter and Hawkes
[1, p. 182]:

Definition 2.3. If 3(p) 1‘ ¢, let cp(c) =n{CG(H/K): H/K
is an 3 central p-chief factor of G}.

We note that CP(G) is a characteristic subgroup of G
such that G/CP(G) E 3(p); hence G

3(p)
Proposition 2.4. Let NdG and 3(p) 99¢. Then

G Cp(C).

CPCN) " N n n[CG(H/K): H/K is an N-3 central p-chief factor of G}.

Proof. Let <1>=HOCH C...CH8=NC:...CHt'-'G be

1
a chief series of G. (If N 8 <1> or G, the result is trivial.)
Refining <1> - Ho C H1 (2...: H8 ' N to a chief series of N, we
see from 2.2 that CPCN) ‘ n{cN(Hi+1/u1): O s i s 8-1 and Hi+lmi
is an N-3 central p-chief factor}. Since N centralizes any
chief factor above N, CPCN) ' H{CN(H1+1/Hi): 0 s i s t-l and
Hi+1/Hi is an N-3 central p-chief factor}. The desired con-
clusion now follows from the Jordan-Hinder Theorem.

Definition 2.5. Let Sp be a Sylow p-complement of G
and N a normal subgroup of G. Let X(Sp,N) = NG(SP n CPCND
if 3(p) 1‘ ¢, and X(SP,N) = Sp if 3(p) - ¢. Let 2 be a set
of Sylow p-complements of G. A relative 3 normalizer (with
respect to N) is the subgroup DNCE) ' n{X(Sp,N): Sp 6 2}.

Also let TNGJ) - n{X(Sp,N): s“ e 2 and p 6 11(3)}.

We note that the various DNG) form a conjugate class of
subgroups of G, and the same is true for the various 'TNGJ).
If N - G, we write D61) in place of D603).

Proposition 2.6. Let 3(p) 1‘ ¢, N 4 G, and Sp a Sylow
p-complement of G. Then X(SP,N) covers the N-3 central
p-chief factors of G and avoids the N-3 eccentric p-chief
factors of G.

Proof. Since 0pm) 4 G, NG(Sp n CP(N)) is a cover-avoid
subgroup which covers the p-chief factor H/K if and only if
Cp(N) C CGCH/K) (Appendix, Theorem A). Every chief factor of
G is G-isomorphic to a chief factor above or below N. Hence
it is sufficient to prove the statement for chief factors above

and below N. If H/K is a p-chief factor with K :2 N,

N/CN(H/K) =1<1>'€ 3(p); i.e. H/K is N-3 central. Since

cpCN) s: N c CGO-l/K), NG(Sp n cpcm) covers H/K. Suppose H/K

is a p-chief factor of c with H s: N. Then NG(SP n cpCN))
covers H/K if and only if CPCN) s: CNm/K)' If CPCN) S: CN(H/K),
H/K is N-3 central since N/CPGN) 6 3(p). If H/K is N-3
central, 2.2 implies 0pm) : cNm/K)° Hence NG(Sp n Cp(N))
covers the p-chief factor H/K if and only if H/K is N-3

central.
Corollary 2.6.1. With the same hypothesis as Proposition

9 B p
2-6. NG(S n CPCND NG(S n 113“»).

Proof. Since N3(p) 4 G and {N3(p) C CP(N):

NG (Sp n Cp(N)) S'- NG(Sp n N Each subgroup clearly covers

3(p))'

all q-chief factors of G with q i p, and NG(Sp n N covers

3(p))

the p-chief factor H/K if and only if N C CN(H/K), if and

303)
only if H/K is N-3 central. Hence the two subgroups have the

same order and are equal.
If 3 I‘n, the formation of nilpotent groups, then

3 = {3(9)} where 3(p) ' [<l>} for all primes p. Hence if

N <1 G, then N N. By 2.6.1 DNG) = n{NG(Sp n N):

(p)
plo(G) and S2 E z} is the relative system normalizer as de-
fined by P. Hall in [3].

Proposition 2.7. let N 4 G, 2 3 {SP: plo(G)} a set of
Sylow p-complements of ‘G, and n a set of primes. Then

S ‘l]{X(Sp,N): Sp 6,: and p 6 n} avoids N-3 eccentric n-chief

factors of G, and covers all other chief factors of G.

Proof. If 3(p) = ,3 with p e 11, s : X(SP,N) = s", which
avoids all p-chief factors of G. (If 3(p) - ¢, all p-chief
factors are N-3 eccentric.) If 3(p) f ¢ and p 6 n,

S : X(SP,N), which avoids all N-3 eccentric p-chief factors

by 2.6. Therefore 8 avoids all N-3 eccentric n-chief factors.
Since [G: X(SP,N)] is a power of p and equals the product

of the orders of the N-3 eccentric p-chief factors in a given
chief series of G, [6:8] = n[[G: X(SP,N)]: p E n} equals the
product of the orders of the N-3 eccentric n-chief factors in

a given chief series of G. By considering the order of S, we
see that 8 must cover all other chief factors of G.

Corollary 2.7.1. Let N 4 G and z a set of Sylow p-
complements of G. Then DN(£) covers N-3 central chief factors
of G and avoids N-3 eccentric chief factors of G. Th(2)
avoids all N-3 eccentric n(3)-chief factors of G and covers
all other chief factors of G.

Proof. In Proposition 2.7, let n be the set of all
primes and 'nC3), respectively.

Proposition 2.8. Let N 4 G and e an epimorphism of
G. Then an N-3 central 0N-3 eccentric) chief factor of G
maps to an Ne-3 central (Ne-3 eccentric) chief factor of G6
or <l>. Conversely, any N9-3 central (Ne-3 eccentric) chief
factor of G8 is the image of an N-3 central (N-3 eccentric)
chief factor of G.

Proof. Both statements follow immediately from 1.3.

Proposition 2.9. Let N 4 G, 2 ' {SP: p|o(G)} a set of

Sylow p-complements of G, n a set of primes, and e an

epimorphism of G. Then (h{X(Sp,N): Sp 6 2 and p 6 fl})6
= n{x<spe,Ne): sp 6 z and p e 11}.

Proof. Let s - n{X(SP,N): sp 6 z and p e n} and
s1 .. n{x(spe,Ne): sp 6 z and p e 11}. If p e n and 3(p) = p.
then 39 : X(SP,N)s = Spe = x<spe,Ne). If p E 11 and 3(p) ’4 ¢,
then so : X(Sp,N)6 - NG(SP n N

p .
)BGNGefls nN )9). since

3(1)) 3(9)

4 G, Sptfl N is a Sylow p-complement of N3(p). Hence

N3(p) 300
p a p g p

89 c NGe(spe n (N9) - x<spe,Ne). and 39 c 3 But by 1.2,

3(p)) 1'

2.7, and 2.8, So and 81 cover and avoid the same chief factors

of Ge. Hence $9 B 81'

Corollary 2.9.1. Let N 4 G, 2 a set of Sylow p-complements
of G, and e an epimorphism of G. Then (DN(2))6 B DNe(ze)
and (Tumm - T119639) where 29 =- {spe: sp 6 s}.

Proof. In Proposition 2.9, let n be the set of all
primes and "(3), respectively.

The cover-avoidance property of 2.7.1 does not in general
characterize DNQ). If N 3G and 3 is the formation of
nilpotent groups, T.0. Hawkes has given an example [5, 11.12,

p. 730] of a group G and a subgroup S of G with the cover-
avoidance property, but 8 is not a system normalizer of G.

PrOposition 2.10. Let N14 G and M.G G such that M
covers all q-chief factors of G with q f p, all N-3 central
p-chief factors of G, and avoids all N-3 eccentric p-chief
factors of G. Then. M contains a Sylow p-complement of G,

P

and if S is any such Sylow p-complement, then M = X(SP,N).

10

Proof. It is clear that ‘M contains a Sylow p-complement
of G. If G has no N-3 central p-chief factors, M is a
Sylow p-complement of G, and so the statement is true in this
case. Assume H/K is an N-3 central p-chief factor of G.
Since N/NOCG(H/K) E 3(p), any p-chief factor above CG(H/K)
is N-3 central. By hypothesis, M. covers H/K and G/CG(H/K).
Hence by 1.1, mM/mM is a p-chief factor of M and
M n CG(H/K) - cManM/mu). Intersecting the terms of a chief
series of G with ‘M, we obtain
Mlfi fl{Cb(H/K): H/K is an N-3 central p-chief factor of G}
= n{CM(mM/mM): H/K is an N-3 central p-chief factor of G}
=f]{GM(A/B: A/B is a p-chief factor of M]
a: Op'pm) [5, 5.4, p. 686]. Now let Sp be a Sylow p-complement
of 6 contained in M. Then MCNG(SP n 0p.p(M))
: NG(SP n op.p(M) n N) - Nccs" n u n open) (by 2.4) = xcspm).
Since M and X(SP,N) have the same order, M = X(SP,N).
Corollary 2.10.1. Suppose DN(2), a relative 3 normalizer
of G, has index a power of p in G. Then if M is any sub-
group of G which covers N-3 central chief factors and avoids
N-3 eccentric chief factors, M. is a relative 3 normalizer.
Proof. Since DNCZ) has index a power of p in G,
DN(2) = X(Sp,N) where Sp 6 2. By 2.7.1, all q-chief factors
of G, q i p, are N-3 central. Proposition 2.10 now implies
the statement.
Proposition 2.11. Let G be a group such that 3 nor-
malizers coincide with 3 projectors. Then if M is any sub-

group which covers 3 central chief factors and avoids 3

ll

eccentric chief factors, M is an 3 normalizer of G.

Proof. Induct on o(G) and let A be a minimal normal
subgroup of G. By 1.2 and 2.8, MA/A has the same cover-avoid
property in G/A. Also by 2.9.1 and Theorem B, 3 projectors of
G/A coincide with 3 normalizers of CIA. Hence MA/A - FA/A
where F is an 3 projector of G. Suppose A is 3 central.
Then M :2 A, F '2 A, and F = M. If A is 3 eccentric, F is
an 3 projector of FA and F n A =- <l>. Hence (FA)3 B A,
and since A is abelian, F is conjugate to M in FA.
(Theorem E.) Therefore M is an 3 projector of G.

Proposition 2.12. Let N 4 G. Then N 6 3 if and only
if N is s 11(3) group and all 11(3)-chief factors of G are
N-3 central. Hence if z is a set of Sylow p-complements of
G, N E 3 if and only if N is a 11(3) group and TNGZ) =- G.

Proof. Suppose N 6 3; than N is a 11(3) group. By
2.2, all chief factors of G below N are N-3 central, and
clearly all 11(3)-chief factors above N are N-3 central.
Conversely, suppose N is a 11(3) group and all 11(3)-chief
factors of G are N-3 central. Again by 2.2, all chief
factors of N are 3 central. Therefore N 6 3. The second
statement now follows readily from 2.7.1.

Corollary 2.12.1. Let N 4 G and )3 a set of Sylow

p-complements of G. Then G = ‘I‘NQIN3

Proof. TN (2:)N3/N3 - TN/N (ms/N3) = G/NS-
Proposition 2.13. Let Z 8 {SP: p|o(G)} be a set of Sylow

P

p-complements of G, and for each Sp 6 2, let Sp 9: T C G.

Then 2 reduces to T - n{Tp: p‘o(G)}. In particular, 2': reduces

12

to DN(2).

Proof. Let Sp 6 )3; then [T: Sp 0 T] = [Tp: Sp], a power of
p. Hence 2 reduces to T. The second statement follows since
X(SP,N) :2 S1) for all primes p.

Proposition 2.14. Let 2 be a set of Sylow p-complements
of G. If D(Z) C M C G and 2 reduces to M, then
D03) C DC: 0 M)-

Proof. If 3(p) =45, D(2) c sp n M for sp 6 2. If
3(p) # ¢, let H/K be an 3 central p-chief factor of G. Then
G/CG(H/K) E 3(p) c 3. Since D03) c: M, M covers G/CG(H/K) and
H/K. By 1.1, HWM/KnM is an 3 central p-chief factor of M,
and canM/mM) - M n cam/K). Therefore CPCM) : M n cp(c).
Since D(2) c M, D03) normalizes CPCM) and Sp 0 CP(G), where
Sp 6 2. Therefore D03) C NM(SP n Cp(M)), and so D03) 5 D03 0 M).

The following proposition was proved by T.0. Hawkes [4, 3.3,
p. 244] in the case that 3 :2 71, the formation of nilpotent groups.
Trivial modifications can easily be given to include the case
3 i 72; however, for completeness we give an alternate proof.

Proposition 2.15. Let D03) be an 3 normalizer of G.
Then there exists an 3 projector F of G such that DOE) c F,
and 2 reduces to F.

Proof. We may suppose G G 3. Let A be minimal normal
in G. Then by induction DG)A/A = DEA/A) : Fl/A where F1/A
is an 3 projector of CIA and EAIA reduces to Fl/A' Suppose
o(A) is a power of p.‘ If q #p and Sqez, Acsq and
[F1: Sq 0 F1] = [Fl/A: Sq/A n F1/A], a power of q. For Sp 6 2,

[F1: Sp n F1] '3 [171: SpAn li'1'_j[(Sp n F1)A: Sp n F1], 3 power of p.

13

Hence 2 reduces to F1, and by 2.14 D03) 1: D0: n F1). If
F1 < G, by induction Dc; 0 F1) 9'. F where F is an 3 pro-
jector of F1 and Eli F1 reduces to F. Hence 2 reduces to
F, F is an 3 projector of G (Theorem B), and D(2) : F.
Suppose F1 3 G. Since G € 3 and CIA E 3, every complement
of A in G is an 3 projector of G (Theorem E). But
G =AD(}:) and A is 3 eccentric. Hence D03) n A =<l> .
and D03) is an 3 projector of G. 2 reduces to DC.) by
2.13.

Proposition 2.16. Suppose G3

an 3 projector of G. Then 3 projectors of G coincide with

is complemented in G by

3 normalizers of G, and all complements of G in G are

3

conjugate in G.

Proof. Suppose G ' GSF, F n G3

projector of G. Let D c F be an 3 normalizer of G; then

-«<l>, and F is an 3

G ‘ 63D and D 3 F. Now suppose H is a complement of G3

in G. Let A : G3 be minimal normal in G. (If G3

the statement is trivial.) Then HA/A complements G /A in

3

CIA. By induction, HA = FA where F is an 3 projector of

-«<1>,

G. If A is 3 central, A c F and so H c F. Therefore

H = F since both are complements of G in G. Suppose A

3
is 3 eccentric, then as in the proof of 2.11, (FA) = A and

3
F is conjugate to H.
Proposition 2.17. Let N be a normal “(3) subgroup of
G. If N3 is complemented in N by an 3 projector of N,
then N3 is complemented in G. Furthermore all complements of

N3 in G are conjugate and are precisely the various TN(Z).

14

Proof. By 2.12.1, G = TN(2)N where 2 is a set of

3

Sylow p-complements of G. Now DNQ) n N C TN(Z) n N , but

8 8
since N is a normal 11(3) subgroup, N c. Sp for all Sp 6 2

such that p 6 11(3)‘. Therefore N n TN (2) C DN (2;) and

TN(2) n N ' DNQ‘.) n N " D0: n N) f] N =<1> by hypothesis

3 3 3

and 2.16. If H is a complement of N in G. N ' N (H n N):

3 3
and HnN is an 3 projector of N by 2.16. Clearly

NG(HnN) 2H, and so NGOinN) -HNNa'ln N) ‘HO-lflN)

(Theorem D) = H. Hence if H and H are complements of

l 2
N3 in G, HI -= NGCH1 n N) is conjugate to H2 8 NG(H2 n N).

Since '1‘N (Z) is a complement of N in G, and the various

3
TN(Z) form a conjugate class, the statement is proved.

Corollary 2.17.1. Let N14 G. If N3 is an abelian
n(3) group, the conclusions of Proposition 2.17 hold.

Proof. Theorem E states that N3 is complemented in
N by an 3 projector of N.

Example 2.17.2. Let G be a solvable Frobenius group
with Frobenius Kernel K. Then K is complemented in any
solvable extension of G, and all such complements are conjugate.

Proof. Let n I {p: p|o(K)} and 3 the saturated forma-
tion of groups which are p-nilpotent for all p 6 n. 3 is
locally defined by: 3(p) = {<1>} if p E n and 3(p) = 3 if
p G 11. Hence 11(3) is the set of all primes. We now show that
if H is any solvable Frobenius group, and the Frobenius Kernel
L of H is a H Hall subgroup of H, then H ' L. Since H/L

3

is a 11' group, H3 : L. If Kl/K2 is any chief factor of H

C '- . = .
with K1 L, then anal/1(2) L Hence HSS L Now let M

15

be a complement of K in G. We show that M is an 3 pro-
jector of G. M E 3 since M is a n' group. Suppose Mt:.U;
then U = (K n U)M, a solvable Frobenius group with kernel K O U.

Therefore ' (K.n U) and U = U M. Proposition 2.17 now implies

"3 3

the conclusion.

Proposition 2.18. Let N14 G; than N E 3 if and only if
N is s 11(3) group and N0 (GHQ (G) 6 3.

Proof. Suppose N is a n(3) group and N§(G)/Q(G) E 3.
Let 2 be a set of Sylow p-complements of G. Then

TN(2)§(G)/<D(G) = )(Z§(G)/§(G)) - clue). Therefore

TN§(G)/t(c
TN@)§(G) = G, and so Tum) = G. By 2.12, N E 3. The converse
is clear.

The following corollary is due to Gaschfitz [5, 3.5, p. 270]
in the case that 3 "n; furthermore, the solvability of G is
not necessary in Gaschfitz result. The case 3 the formation of
supersolvable groups, for example, does not seem to appear in the
literature.

Corollary 2.18.1. Let D,M 4 G with M a n(3) group
and D C §(G). If M/D E 3, then M.€ 3.

Proof. Suppose MID 6 3. Then M§(G)/§(G) = M§(G)/D§(G)

: M/DOޤ(G)) 6 3. Hence M E 3 by 2.18.

Corollary 2.18.2. Suppose 3 a‘n, and 3 is normal sub-
group closed. If 3* is any non-empty formation,

33* . {6:3 K <1 G with K e 3 and G/K e 3*} is a saturated
formation.

Proof. Since 3 is normal subgroup closed, G E 33* if

*
andonly if 0*63. Let 6633 and N4G; (G/N)*
3

16

= c ,N/N e 3. If c/Nl and c/N2 e 33*, c *Ni/Ni e 3, i = 1,2.
3
Hence (G/NIONZ) * - c *ch n szl n N2 ‘1' a */(N1 n N2)OG * e 3.
3 S 3
If c/uc) e 33*, then a pan/Ms) e 3. By 2.18, c * e 3.
3

Proposition 2.19. Let N14 G. Then N 6 3 if and only
if N is a 11(3) group and all 11(3)-chief factors of G be-
tween §(G) and Fit(G) are N-3 central.

Proof. Assume the criteria. By 2.8 and 2.18, we may
assume §(G) "<1>3 Fit(G) is a completely reducible G-module
[5, 4.5, p. 279], say Fit(G) -=CE{H1: l s i S n} where the Hi
are minimal normal subgroups of G. We may suppose
CE[Hi:
is a 11(3)‘ group, for if N = <l>, there is nothing to prove.

lSiss};szl, isa 11(3) group and @{Hi: s+lsi$n}

By hypothesis, N/CN(H1) E 3(p) : 3 for l s i s s and appropriate
primes p. Therefore Nfln{CN(H1): l s i s s} E 3. Since N is
a normal n(3) group, N centralizes CE{Hi: 8+1 5 i s n}. Hence
n{CN(Hi): l s i s s} c N n CG(Fit(G)) " N n Fit(G), and so
N/N n Fit(G) E 3. Let H/K be a n(3)-chief factor above Fit(G).
If H.fi N 3 K.n N, N C CG(H/K) and H/K is N-3 central. If
H n N 1‘ K n N, H/K is G-isomorphic to H n N/K n N. By 2.2, H/K
is N-3 central if and only if every chief factor of N between
K.n N and H n N is 3 central. Since K.n N 2 Fit(G) H N
and N/N fl Fit(G) E 3, H/K is N-3 central. Therefore all
fl(3)-chief factors of G are N-3 central, and N E 3 by 2.12.
Also the converse is clear by 2.12.

Definition 2.20. Let N 4 G. A p-maximal subgroup M of
c is N-3 normal if M n N/Core(M) n N e 3(p), and is N-3

abnormal otherwise.

17

Proposition 2.21. Let N 4G and M a p-maximal subgroup
of G. Then M is N-3 normal if and only if M complements an
N-3 central p-chief factor of G.

Proof. Suppose M is N-3 normal; then 3(p) 1‘ ¢. Let
K = Core (M). Then G/K has a unique minimal normal subgroup H/K,
CG(H/K) = H, and M complements H/K [5, 3.2, p. 159]. If
N n H c M, N n H s: x, and H/K is N-3 central. Suppose
N n H ¢ M, then G = (N n H)M and N 3 (N n H)(Mfl N). Therefore
N/cNai/K) =N/N n H 'IMn N/Mn Hn N - Mn N/Kn N e 3(p). Suppose
M complements the N-3 central p-chief factor H/K. Let
c -= CGG-l/K). Then Core(M) = c n M and c = H(C n M). Hence
C/C n M is G-isomorphic to H/K, and M complements C/C n M.
Therefore we may suppose CG-(H/K) - H and K = Core (M). If
N n H i: M, than N n H c K, and so N s: Cam/K) = H. Therefore
N =N n H c M and Mn N/CoreCM) n N -<l> e 3(p). (3(p) 1‘ ¢
since H/K is N-3 central.) If N n H ¢ M, G = (N n H)M and
N - (N n H)(M n N). Hence Mn N/Kn N ‘-'- N/N n H - N/CNOI/K) e 3(p).

Proposition 2.22. Let N 4 G. Then N E 3 if and only
if N is a 11(3) group and for p 6 11(3) with 3(p) C 3 all
p-maximal subgroups of G are N-3 normal.

Proof. Assume the criteria and induct on o(G). We may
assume N 9‘ <l>. Let A be a minimal normal subgroup of G
contained in N. By 2.8, 2.21, and the induction hypothesis,

N/AE3. Then N CA. Suppose N =A and A is ap-chief

3 3
factor of G. By 2.17.1, TNGS) is a p-maximal subgroup of G
complementing A, where 2 is a set of Sylow p-complements of G.

If 3(p) C 3, Tum) is N-3 normal by hypothesis. If 3(p) = 3,

18

TN(2) n N *3 DCE. n N) E 3 (Theorem F). Hence, in any case TN (2:)
is N-3 normal. By 2.21, A is an N-3 central 11(3)-chief
factor. But A n TN(2) '- <l>, a contradiction to 2.7.1. Hence

N3 = <l> and N E 3. Conversely, if N 6 3, all 11(3)-chief
factors of G are N-3 central. Hence all 11(3)-maximal sub-
groups of G are N-3 normal by 2.21.

Example 2.22.1. Let N 4G and 11 a set of primes. Then
N is p-nilpotent for all p E 11 if and only if N n M C Core(M)
for all n-maximal subgroups M of G.

Proof. Let 3 be the formation of groups which are p-
nilpotent for all p E n. Using the local definition of 3
given in 2.17.2, we obtain the statement as an immediate consequence
of 2.22.

Proposition 2.23. Let 2 be a set of Sylow p-complements

of G. Then D-DQZ) is an 3 projector of NGOZOGs).

B. Fischer has defined 3 normalizers of G to be 3

projectors of NGG 0 CS) in the case that 3 :2 71 [2, 8.4, p. 63].

Proof. First note that D S'- NGQ". 11 G3): ‘ If 3(p) '3 ¢,

p ) for Sp 6 2; if 3(P) 9‘ ¢, NG(SP n G

s cNG(spn 6

gm)

Now induct on 0(6) and let

3

3) since G3: G3“).

A be minimal normal in G. Then DA/A is an 3 projector of

P
:
NG(S n G

P
NG/AazA/A n GSA/A). But NG (z: n GS)A/A s: NG/A((s n c3)A/A)
‘3 NG/A(SpA/A n G3A/A) for all Sp 6 2. Therefore DA/A is an

3 projector of NGQGG )A/A. If A is 3 central, then

3

D 2 A, and D is an 3 projector of N60: 0 G ) by Theorems

3

B and F. If G ¢ CG(A), NG(2 n G ) avoids A, and the natural

3

isomorphism from NG (2 n G

S

)A/A to NGan G) maps DA/A onto

3 3

19

D since D G NGQ". n G ). Hence we may assume A is 3 eccentric

3

and G C CG(A). Therefore G *3 CG(A)D, and so A is minimal

3
normal in AD. Now G/CG(A) F7 AD/CAD(A) é 3(p), but AD/A '3' D E 3.

Hence D is an 3 projector of AD. Again by Theorem B, we

conclude that D is an 3 projector of NGQZO G

)0
3
Corollary 2.23.1. Let 2 be a set of Sylow p-complements

of G. Then D03) is an 3 projector and an 3 normalizer of
NGQ‘. n as). If c e 3, D03) c: NCO: n G3) and

G = <NG(2 n G8)8: g E G>. Hence every finite solvable group is
generated by subgroups in which 3 projectors coincide with 3

normalizers .

Proof. By 2.23, D(2) is an 3 projector of N60: n G ).

3

By 2.13 2 reduces to NG(Zn G ), and by 2.14 13(2)

3
: D(z n NG(2 n 63)). But since D(Z) is a maximal 3 subgroup,

we must have equality (Theorem F). Now if D(2) covers the p-

chief factor H/K of G, G G G : CGCH/K), and so

3 8(9)

NGQZ n G ) covers H/K. If G G 3, <1>C G and D(2) avoids

3 3
Git/63' Hence 0CD(2)) < OCNGG: n GS)). If N66: (1 GS) is con-

tained in a proper normal subgroup M of G, we may suppose M

is maximal normal. But G3 5 G = CG(G/M), and so

G = MNG(2 0 G3) = M, a contradiction. Hence G = (NCO: n G )8:

S3
g€G>.

Corollary 2.23.2. Let F be an 3 projector of G. Then
F is an 3 normalizer of G if and only if F normalizes a

Sylow p-complement of G for all primes p.

3

Proof. Suppose F = D03); then F C NGQ'J n G ). Conversely,

3

if F G NGOZ‘. n G ) for a set 2'. of Sylow p-complements, then F

3

20

is an 3 projector of NG(2 n G ) and so is conjugate to D(Z).

8

Proposition 2.24. Let 72 C 3. If G = NF, N 4 G, F 4 G,
N672 and F63, then G63.

Proof. Induct on o(G). The hypothesis is preserved under
epimorphic images. Hence there exists a unique minimal normal
subgroup A and CIA E 3. If G 1 3, A is complemented in G
by Theorem E. Hence A is self-centralizing. Since A G N,

A : Z(N) and so A = N. (N # <l> since G 6 3.) Therefore
FnN=N and G'FES, or FnN'<l> and G8NE3.

Proposition 2.25. Let 71 C 3. Suppose G = AB = AC B BC
with A,B abelian subgroups of G and C 6 3, then G 6 3.

Proof. Induct on o(G). The hypothesis is preserved under
epimorphic images; hence without loss of generality, there exists
a unique minimal normal subgroup M of G and G/M E 3. If
G 1 3,‘M is complemented in G and is thus self-centralizing.
Since G = AB with A,B abelian, Core(A) fi<l>' or Core(B) # <1)
[7, 13.3.3, p. 384]. Hence we may suppose ‘M c A. Therefore
M = A 3 GS. If B and C are both proper subgroups of G, they
are complements of 'M 8 G3. Hence B is conjugate to C (Theorem
E), but G = BC, a contradiction. Therefore G = B or G = C.

In either case, G E 3.

0.H. Kegel proved the following proposition in [6] for the
case 3 the formation of supersolvable groups. The proof given
here is very similar to Kegel's proof.

Proposition 2.26. Let ‘n c 3. Suppose G = AB = AC = BC

with A,B E‘fl, C a Sylow tower group, and C E 3. Then G 6 3.

21

Proof. Suppose not, and let G be a counterexample of

minimal order. As in 2.25 G is the unique minimal normal sub-

3

group of G, and G is self-centralizing. Let p be a prime

3

such that C has a non-trivial normal Sylow p-subgroup. Let
AP, Bp, and CP be the normal Sylow p-subgroups of A, B, C,

respectively. Then P =- A B -= A C = B C is a normal Sylow
P P P P P P

p-subgroup of G [6, 1, p. 43]. Hence GS: P. Since CG(G

=P.NowA BCZG andsoAnB=P or
P”? () P P

3)
= GS: G3

Ap n B1) = <1>. If Ap 0 BP '= P '3 G , G is an 3 central chief

3 3
factor, which is impossible. Hence AP n Bp = <1>. Suppose

A =<1> then B =IG andso B IB‘G. Since A CCG
p 9 p 3 p 3 ’ 3

both A and C are complements of B = G By Theorem E, A

3.
is conjugate to C, an impossibility since G = AC. Therefore
Ap ’9 <l>, and likewise B1) 1‘ <1>. Let K be an A-invariant

complement of Ap in G . Then there exists an A-invariant

3

subgroup T of GS of index p in G3.

the product of K and a subgroup of index p in AP.) Suppose

(T can be taken as

o(Bp)2p2. Then Tani<1> and TanCG But

8.
(T n up)“ - ('1‘ n 3‘)“ c 1: c: (:3. Hence o(Bp) = p and likewise
o(Ap) = p. Let AP, be the complement of AP in A. Then

C

A (K) = <l>, for otherwise G
I

p 3

Hence AP. is represented faithfully on K, and so AP, is

= APK is not self-centralizing.

cyclic. Therefore A is cyclic, and likewise B is cyclic.

Hence G E 3 by 2.25.

BIB LIOGRAPHY

BIBLIOGRAPHY

R. W. Carter and T. 0. Hawkes, The 3-normalizers of a finite
soluble group, J. Algebra 5 (1967), 175-202.

B. Fischer, Classes of conjugate subgroups in finite solvable
groups, Lecture Notes, Yale University, 1966.

P. Hall, 0n the system normalizers of a finite soluble group,
Proc. London Math. Soc. 43 (1937), 316-323.

T. 0. Hawkes, 0n formation subgroups of a finite soluble
group, J. London'Math. Soc. 44 (1969), 243-251.

B. Huppert, Endliche Gruppen I, Die Grundlehren der
mathematischen.Wissenschaften in Einzeldarstellungen 134,
Springer-Verlag, Berlin - Heidelberg, 1967.

0. H. Kegel, Zur Struktur mehrfach faktorisierbarer endlicher
Gruppen, Math. Z. 87 (1965), 42-48.

W.R. Scott, Group Theory, Prentice-Hall, Inc., Englewood
Cliffs, 1964.

22

APPH‘IDIX

APPENDIX

For the convenience of the reader, we State and prove a
well known result of P. Hall, which is contained in Theorem 7.2
of [3].

Theorem A. Let N‘d G, SP a Sylow p-complement of G,
and H/K a p-chief factor of G. Then:
1). If NG(Sp) does not avoid H/K, CGO-I/K) = c.
11). If N ¢ ch/K), NG(SP n N) avoids H/K.
111). A If N : CGO-I/K), NG(SP n N) covers H/K.

Proof. i). Suppose NG(SP) does not avoid H/K. Then

in G/K, N (SpK/K) does not avoid H/K. Hence we may assume

G/K
K = <1>. Let h e H n NG(SP), then sp 5: Cam“ Let H1 - [mo]

and P a Sylow p-subgroup of G. Then H14 G and H1 3 [h,P]<: H.

Therefore H ..<1> and h 6 Z(G)s Hence <h> = H C Z(G).

1
ii). Suppose NG(Sp n N) does not avoid H/K. Again we may

suppose K -'<1>. If H,n N =1<1>, then [H,N] =<1‘> and
N C CG(H). Suppose H C N. Then H.n NN(Sp n N) # <l>, and so
NN(Sp n N) does not avoid some p-chief factor of N below H.

By 1), this chief factor is central in N, and by Proposition 2.2

(with 3(p) = {<1>}). N c com).

111). Suppose Ncch/K).. Let G =HN. Then NG(SpnN) 2

1

NG (81) n (:1), and H/K : Z(GIIK) c N ((sp n Gl)l(/K) .

1 Gllx
N (Sp n G )K/K. Therefore H G N (Splfi G )K C N (Sp n N)K
G1 1 G1 1 G1
: NG(sp n N)K.
23

24

Theorem B [5, 7.9, p. 699]. a). If U c c and F is
an 3 projector of G with F c U, then F is an 3 projector
of U.

b). If N 4 G and F is an 3 projector of G, then FN/N
is an 3 projector of G/N.

c). If F1/N is an 3 projector of G/N and F is an 3
projector of F1, than F is an 3 projector of G.

Theorem C [5, 7.10-b, p. 700]. Every solvable group G
possesses 3 projectors, and all 3 projectors of G are
conjugate in G.

The following theorem is contained in the proof of [5,
7.11, p. 701]:

Theorem D. Let G be a n(3) group. If F is an 3
projector of G and F C U C G, then NG(U) 3 U. In particular,
F is self normalizing in G.

Theorem E [5, 7.15, p. 703]. If G is abeIian, then the

3

3 projectors of G are precisely the complements of G3 in G.
The following theorem is contained in the proof of [1,
4.1, p. 185]:
Theorem F. Let 2 be a set of Sylow p-complements of G

and D(£) an 3 normalizer of G. Then D(Z) 6 3.