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The Classification of Quasithin Groups: II. Main Theorems: The Classification of Simple QTKE-groups
 
Michael Aschbacher California Institute of Technology, Pasadena, CA
Stephen D. Smith University of Illinois at Chicago, Chicago, IL
The Classification of Quasithin Groups
The Classification of Quasithin Groups
Hardcover ISBN:  978-0-8218-3411-4
Product Code:  SURV/112
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1339-2
Product Code:  SURV/112.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-0-8218-3411-4
eBook: ISBN:  978-1-4704-1339-2
Product Code:  SURV/112.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
The Classification of Quasithin Groups
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The Classification of Quasithin Groups
The Classification of Quasithin Groups: II. Main Theorems: The Classification of Simple QTKE-groups
Michael Aschbacher California Institute of Technology, Pasadena, CA
Stephen D. Smith University of Illinois at Chicago, Chicago, IL
Hardcover ISBN:  978-0-8218-3411-4
Product Code:  SURV/112
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1339-2
Product Code:  SURV/112.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-0-8218-3411-4
eBook ISBN:  978-1-4704-1339-2
Product Code:  SURV/112.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 1122004; 743 pp
    MSC: Primary 20;

    Around 1980, G. Mason announced the classification of a certain subclass of an important class of finite simple groups known as "quasithin groups". The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. Unfortunately Mason neither completed nor published his work. In the Main Theorem of this two-part book (Volumes 111 and 112 in the AMS series, Mathematical Surveys and Monographs) the authors provide a proof of a stronger theorem classifying a larger class of groups, which is independent of Mason's arguments. In particular, this allows the authors to close this last remaining gap in the proof of the classification of all finite simple groups.

    An important corollary of the Main Theorem provides a bridge to the program of Gorenstein, Lyons, and Solomon (Volume 40 in the AMS series, Mathematical Surveys and Monographs) which seeks to give a new, simplified proof of the classification of the finite simple groups.

    Part I (Volume 111) contains results which are used in the proof of the Main Theorem. Some of the results are known and fairly general, but their proofs are scattered throughout the literature; others are more specialized and are proved here for the first time.

    Part II of the work (the current volume) contains the proof of the Main Theorem, and the proof of the corollary classifying quasithin groups of even type.

    The book is suitable for graduate students and researchers interested in the theory of finite groups.

    Readership

    Graduate students and research mathematicians interested in the theory of finite groups.

  • Table of Contents
     
     
    • Part 1. Structure of QTKE-groups and the main case division
    • 1. Structure and intersection properties of 2-locals
    • 2. Classifying the groups with $|\mathcal {M}(T)|= 1$
    • 3. Determining the cases for $L \in \mathcal {L}^*_f(G,T)$
    • 4. Pushing up in QTKE-groups
    • Part 2. The treatment of the generic case
    • 5. The generic case: $L_2(2^n)$ in $\mathcal {L}_f$ and $n(H) > 1$
    • 6. Reducing ${\bf L_2(2^n)}$ to ${\bf n = 2}$ and V orthogonal
    • Part 3. Modules which are not FF-modules
    • 7. Eliminating cases corresponding to no shadow
    • 8. Eliminating shadows and characterizing the ${\bf J_4}$ example
    • 9. Eliminating $\Omega ^+_4(2^n)$ on its orthogonal module
    • Part 4. Pairs in the FSU over ${\bf F}_{2^n}$ for $n > 1$.
    • 10. The case $L \in \mathcal {L}^*_f(G,T)$ not normal in $M$
    • 11. Elimination of ${\bf L_3(2^n)}$, ${\bf Sp_4(2^n)}$, and ${\bf G_2(2^n)}$ for ${\bf n > 1}$
    • Part 5. Groups over ${\bf F}_2$
    • 12. Larger groups over ${\bf F_2}$ in $\mathcal {L}^*_f(G,T)$
    • 13. Mid-size groups over ${\bf F_2}$
    • 14. ${\bf L_3(2)}$ in the FSU, and ${\bf L_2(2)}$ when ${\bf \mathcal {L}_f(G,T)}$ is empty
    • Part 6. The case $\mathcal {L}_f(G,T)$ empty
    • 15. The case ${\bf \mathcal {L}_f(G,T)} = \emptyset $
    • Part 7. The Even Type Theorem
    • 16. Quasithin groups of even type but not even characteristic
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1122004; 743 pp
MSC: Primary 20;

Around 1980, G. Mason announced the classification of a certain subclass of an important class of finite simple groups known as "quasithin groups". The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. Unfortunately Mason neither completed nor published his work. In the Main Theorem of this two-part book (Volumes 111 and 112 in the AMS series, Mathematical Surveys and Monographs) the authors provide a proof of a stronger theorem classifying a larger class of groups, which is independent of Mason's arguments. In particular, this allows the authors to close this last remaining gap in the proof of the classification of all finite simple groups.

An important corollary of the Main Theorem provides a bridge to the program of Gorenstein, Lyons, and Solomon (Volume 40 in the AMS series, Mathematical Surveys and Monographs) which seeks to give a new, simplified proof of the classification of the finite simple groups.

Part I (Volume 111) contains results which are used in the proof of the Main Theorem. Some of the results are known and fairly general, but their proofs are scattered throughout the literature; others are more specialized and are proved here for the first time.

Part II of the work (the current volume) contains the proof of the Main Theorem, and the proof of the corollary classifying quasithin groups of even type.

The book is suitable for graduate students and researchers interested in the theory of finite groups.

Readership

Graduate students and research mathematicians interested in the theory of finite groups.

  • Part 1. Structure of QTKE-groups and the main case division
  • 1. Structure and intersection properties of 2-locals
  • 2. Classifying the groups with $|\mathcal {M}(T)|= 1$
  • 3. Determining the cases for $L \in \mathcal {L}^*_f(G,T)$
  • 4. Pushing up in QTKE-groups
  • Part 2. The treatment of the generic case
  • 5. The generic case: $L_2(2^n)$ in $\mathcal {L}_f$ and $n(H) > 1$
  • 6. Reducing ${\bf L_2(2^n)}$ to ${\bf n = 2}$ and V orthogonal
  • Part 3. Modules which are not FF-modules
  • 7. Eliminating cases corresponding to no shadow
  • 8. Eliminating shadows and characterizing the ${\bf J_4}$ example
  • 9. Eliminating $\Omega ^+_4(2^n)$ on its orthogonal module
  • Part 4. Pairs in the FSU over ${\bf F}_{2^n}$ for $n > 1$.
  • 10. The case $L \in \mathcal {L}^*_f(G,T)$ not normal in $M$
  • 11. Elimination of ${\bf L_3(2^n)}$, ${\bf Sp_4(2^n)}$, and ${\bf G_2(2^n)}$ for ${\bf n > 1}$
  • Part 5. Groups over ${\bf F}_2$
  • 12. Larger groups over ${\bf F_2}$ in $\mathcal {L}^*_f(G,T)$
  • 13. Mid-size groups over ${\bf F_2}$
  • 14. ${\bf L_3(2)}$ in the FSU, and ${\bf L_2(2)}$ when ${\bf \mathcal {L}_f(G,T)}$ is empty
  • Part 6. The case $\mathcal {L}_f(G,T)$ empty
  • 15. The case ${\bf \mathcal {L}_f(G,T)} = \emptyset $
  • Part 7. The Even Type Theorem
  • 16. Quasithin groups of even type but not even characteristic
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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