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  • SATURATED FORMATIONS, SCHUNCK CLASSES

    AND THE STRUCTURE OF FINITE GROUPS

    Abdulmotalab M.O. Ihwil

    B.Sc., A L - Fateh Univers i ty , T r i p o l i , Libya, 1979

    A THESIS SUBb4ITTED I N PARTIAL FULFILLMENT OF

    THE REQUIREMENTS FOR THE DEGREE OF

    MASTER OF SCIENCE

    i n t h e Department

    0 f

    Mathematics and S t a t i s t i c s

    @ Abdulmotalab M . O . Ihwi l , 1986

    SIMON FRASER UNIVERSITY

    Apr i l 1986

    A l l r i g h t s reserved. This t h e s i s mhy not be reproduced i n whole o r i n p a r t , by photocopy

    o r o t h e r means, without permission of t h e au thor .

  • APPROVAL

    Name :

    Degree:

    T i t l e of Thesis :

    Abdulmotalab M.O. Ihwil

    Master of Science

    Saturated formations, Schunck c l a s s e s and t h e s t r u c t u r e

    of f i n i t e groups.

    Chairman : . C . Vi l l egas

    n Dr, N . R . Reilly

    ,dl " I /r Ti. A . R . Freedman Externa l Examiner

    Assoc ia te Professor Department of Mathematics and S t a t i s t i c s

    Simon F rase r Un ive r s i t y

    Date approved: A p r i l 3 0 1 lgg6

  • PARTIAL COPYRIGHT LICENSE

    I hereby grant t o Simon Fraser Un ive rs i t y t h e r i g h t t o lend

    my thes i s , proJect o r extended ossay ( the t i t l e o f which i s shown below)

    t o users o f t he Simon Fraser Un ive rs i t y L ibrary, and t o make p a r t i a l o r

    s i n g l e copies on ly f o r such users o r i n response t o s request from t h e

    l i b r a r y o f any o the r univers iyy, o r o the r educational I n s t i t u t i o n , on

    I i t s own behalf o r f o r one o f i t s users. I f u r t h e r agree t h a t permission f o r m u l t i p l e copying o f t h i s work f o r scho lar ly purposes may be granted

    by me o r the Dean o f Graduate Studies. I t i s understood t h a t copying

    o r p u b l i c a t i o n o f t h i s work f o r f l n a n c l a l gain s h a l l no t be al lowed

    wi thout my w r i t t e n permission.

    T i t l e of Thes i s/Project/Extended Essay

    Author:

    ( s igna tu re )

    tnlah P f . r ) . T M l

    ( name

    APRIL 30,1986

    (da te)

  • Abstrac t

    This i s a s tudy of t h e s t r u c t u r e of f i n i t e groups from t h e s tandpoint

    of c e r t a i n c l a s s e s of groups. Examples of such c l a s s e s a r e formations,

    s a t u r a t e d formations and Schunck c l a s s e s . Genera l iza t ions of Hal l

    subgroups a r e t h e dominating theme, and we begin wi th t h e theorems of Hal l

    and Car t e r about t h e ex i s t ence and conjugacy of Hal l and Car t e r subgroups

    ( r e spec t ive ly ) i n f i n i t e so lvab le groups. i

    A b a s i c no t ion i n r e c e n t work i s t h a t of covering subgroups of a

    f i n i t e group, where t h e main r e s u l t i s t h a t if f i s a s a t u r a t e d formation

    then every f i n i t e so lvab le group has f-covering subgroups and any two of

    them a r e conjugate.

    A more genera l no t ion than t h a t of covering subgroups i s t h a t of

    p r o j e c t o r s ; however, i n t h e case of s a t u r a t e d formations t h e covering

    subgroups and p r o j e c t o r s of any f i n i t e so lvab le group coinc ide and form

    a s i n g l e conjugacy c l a s s . Moreover, t h e covering subgroups (p ro jec to r s )

    f o r t h e formation of f i n i t e n i l p o t e n t groups a r e t h e C a r t e r subgroups.

    In a d d i t i o n , pul l-backs exis . t f o r the f -pro jec . tors a s soc ia t ed with a

    s a t u r a t e d formation f of f i n i t e so lvab le groups, and y i e l d s t h e

    cons t ruc t ion of c e r t a i n formations.

    A n a t u r a l ques t ion i s : which c l a s s e s F g i v e r i s e t o f - p r o j e c t o r s ?

    The answer i s t h a t f o r Schunck c l a s s , which a r e more genera l than

    s a t u r a t e d formations, every so lvab le group has &covering subgroups iff F

    i s a Schunck c l a s s . In t h i s case F-covering subgroups a r e conjugate.

    ( i i i )

  • This i s now known t o be t r u e i f "F-covering subgroups" i s replaced by

    ''F-proj ectors", and we prove t h i s r e s u l t using p r o j e c t o r s .

    In f a c t these theorems can be extended t o f i n i t e IT-solvable groups,

    and i f F i s a IT-saturated formation (or a IT-Schunck c l a s s ) then every

    f i n i t e IT-solvable group has f-covering subgroups (p ro jec to r s ) and any

    two of them a r e conjugate.

    The ex i s t ence of p r o j e c t o r s i n any f i n i t e group i s proved; however,

    they may no t be conjugate and may no t coincide with t h e covering subgroups,

    which might not a t a l l e x i s t , i n t h i s ca&. But i f F is a Schunck c l a s s

    then F-projec tors and F-covering subgroups do coincide i n groups i n UF,

    although t h e F-pro j e c t o r s need no t be con jugate .

  • Acknowledgement

    I wish t o thank my s e n i o r supervisor Professor J . L . Berggren f o r

    suggest ing the t o p i c and f o r h i s s i n c e r e and f r i e n d l y h e l p and guidance.

    I would a l s o l i k e t o thank Professor N.R. R e i l l y and Professor C . Godsil

    f o r t h e i r va luab le remarks and comments. I wish a l s o t o thank a l l

    f a c u l t y and s t a f f members of the Mathematics and S t a t i s t i c s Department.

    My thanks a l s o t o Cindy L i s t e r f o r typing t h i s manuscript .

  • NOTAT ION

    P \ 9

    < 1>

    1

    Aut (G)

    A \ B

    g i s an element of G .

    elements of groups.

    s e t s , groups.

    c l a s s e s of groups.

    H i s isomorphic with G .

    H i s a subgroup of G .

    H i s a proper subgroup of G .

    H is a normal subgro y p o f G .

    t h e order of t h e group G .

    Index of t h e subgroup H i n t h e group G .

    p d i v i d e s q.

    t h e i d e n t i t y subgroup.

    t h e i d e n t i t y element of a group.

    t h e group of automorphisms of G .

    = ( x : x E A and X B B).

    = ( p : P i s a prime and p \ \ G I ) .

    = g-lhg where g,h C G and G i s a group.

    -1 -1 = x y xy.

    = .

    = ( g E G : hg = h f o r a l l h E G I - t h e c e n t r e of G .

    = ( g E G : hg = h f o r a l l . h E H) - t h e c e n t r a l i z e r of

    PI i n G .

    = ( g E G : hg E H f o r a l l 'h E H I - t h e normalizer of

    H i n G .

  • a group with every element having order a power of t h e

    prime p.

    a group with every element having order a power of p ,

    where p € n, and n i s a f ixed s e t of primes.

    a group with every element having order a power of p ,

    where p is a prime not i n n , n is a f ixed s e t of

    primes.

    on (G) t h e l a r g e s t normal n-subgroup of G . H x K d i r e c t product of H and K .

    Maximal subgroup : proper subgroup, n o 6 contained i n any g r e a t e r proper

    subgroup.

    Minimal normal subgroup of G : normal subgroup M # of G which does

    no t conta in normal subgroups of G except and M.

    ( v i i )

  • TABLE OF CONTENTS

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approval

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstrac t . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . Tab1 e of Contents

    L i s t o f F i g u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1.

    Chapter 2 . P ro jec to r s and Formations . . . . . . . . . . . . . . . Chapter 3 . Pro jec to r s and S c h u n c k ~ a s s e s . . . . . . . . . . . . . Chapter 4. Covering Subgroups and Pro jec to r s i n F i n i t e

    n-solvableGroups . . . . . . . . . . . . . . . . . . . Chapter 5. P ro jec to r s of F i n i t e Groups . . . . . . . . . . . . . .

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography

    . ( i i )

    ( i i i )

    (v)

    . (v i ) ( v i i i )

    ( ix )

    1

    26

    . 58

    ( v i i i )

  • LIST OF FIGURES

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 1 53

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2 61

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 3 64

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 4 66

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 5 67 F i g u r e 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .