Finite Groups Whose 2-Subgroups Are Generated by at Most 4 Elements.
Gorenstein, Daniel.
Finite Groups Whose 2-Subgroups Are Generated by at Most 4 Elements. - 1st ed. - 1 online resource (473 pages) - Memoirs of the American Mathematical Society ; v.1 . - Memoirs of the American Mathematical Society .
Intro -- TABLE OF CONTENTS -- INTRODUCTION -- PART I: SOLVABLE 2-LOCAL SUBGROUPS -- 1. Introduction -- 2. The minimal counterexample -- 3. Odd order groups acting on 2-groups -- 4. The local subgroups of G -- 5. The structure of O[sub(2)(M) -- 6. The case C[sub(R)](B) / 1 -- 7. Proof of Theorem A -- PART II: 2-CONSTRAINED 2-LOCAL SUBGROUPS -- 1. Introduction -- 2. The automorphism groups of certain 2-groups -- 3. Theorem B, the GL(3,2) case -- 4. Theorem B, the A[sub(5)]case -- 5. Theorems C and D, initial reduction -- 6. Theorems C and D, the A[sub(5)] case -- 7. Theorems C and D, the GL(3,2) case -- PART III: NON 2-CONSTRAINED CENTRALIZERS OF INVOLUTIONS -- SOME SPECIAL CASES -- 1. Introduction -- 2. Theorem A -- 3. The Ŝz(8) case -- 4. The Â[sub(n) case -- 5. The M[sub(l2)] case -- 6. Some lemmas -- 7. The SL(4,q), SU(4,q), Sp(4,q) cases -- 8. The direct product case -- 9. The central product case -- PART IV: A CHARACTERIZATION OF THE GROUP D[sup(2)sub(4)](3) -- 1. Introduction -- 2. Preliminary lemmas -- 3. The centralizer of a central involution -- 4. The intersection of W and its conjugates -- 5. The normal four subgroup case -- 6. The cyclic case -- 7. The maximal class case -- PART V: CENTRAL INVOLUTIONS WITH NON 2-CONSTRAINED CENTRALIZERS -- 1. Introduction -- 2. Initial reductions -- 3. Theorem A -- the wreathed case -- 4. Preliminary results -- 5. Maximal elementary abelian 2-subgroups -- 6. Fusion of involutions -- 7. Theorem A -- the dihedral and quasi-dihedral cases -- PART VI: A CHARACTERIZATION OF THE GROUP M[sub(12)] -- 1. Introduction -- 2. 2-groups and their automorphism groups -- 3. Some 2-groups associated with Aut(Z[sub(4)] x Z[sub(4)]) -- 4. Initial reductions -- 5. Elimination of the rank 3 case -- 6. The major reduction -- 7. The non-dihedral case -- 8. The noncyclic case -- 9. The structure of O[sub(2)](M). 10. The structure of S.
9780821899472
Finite groups.
Electronic books.
QA177 -- .G67 1974eb
510/.8 s;512/.2
Finite Groups Whose 2-Subgroups Are Generated by at Most 4 Elements. - 1st ed. - 1 online resource (473 pages) - Memoirs of the American Mathematical Society ; v.1 . - Memoirs of the American Mathematical Society .
Intro -- TABLE OF CONTENTS -- INTRODUCTION -- PART I: SOLVABLE 2-LOCAL SUBGROUPS -- 1. Introduction -- 2. The minimal counterexample -- 3. Odd order groups acting on 2-groups -- 4. The local subgroups of G -- 5. The structure of O[sub(2)(M) -- 6. The case C[sub(R)](B) / 1 -- 7. Proof of Theorem A -- PART II: 2-CONSTRAINED 2-LOCAL SUBGROUPS -- 1. Introduction -- 2. The automorphism groups of certain 2-groups -- 3. Theorem B, the GL(3,2) case -- 4. Theorem B, the A[sub(5)]case -- 5. Theorems C and D, initial reduction -- 6. Theorems C and D, the A[sub(5)] case -- 7. Theorems C and D, the GL(3,2) case -- PART III: NON 2-CONSTRAINED CENTRALIZERS OF INVOLUTIONS -- SOME SPECIAL CASES -- 1. Introduction -- 2. Theorem A -- 3. The Ŝz(8) case -- 4. The Â[sub(n) case -- 5. The M[sub(l2)] case -- 6. Some lemmas -- 7. The SL(4,q), SU(4,q), Sp(4,q) cases -- 8. The direct product case -- 9. The central product case -- PART IV: A CHARACTERIZATION OF THE GROUP D[sup(2)sub(4)](3) -- 1. Introduction -- 2. Preliminary lemmas -- 3. The centralizer of a central involution -- 4. The intersection of W and its conjugates -- 5. The normal four subgroup case -- 6. The cyclic case -- 7. The maximal class case -- PART V: CENTRAL INVOLUTIONS WITH NON 2-CONSTRAINED CENTRALIZERS -- 1. Introduction -- 2. Initial reductions -- 3. Theorem A -- the wreathed case -- 4. Preliminary results -- 5. Maximal elementary abelian 2-subgroups -- 6. Fusion of involutions -- 7. Theorem A -- the dihedral and quasi-dihedral cases -- PART VI: A CHARACTERIZATION OF THE GROUP M[sub(12)] -- 1. Introduction -- 2. 2-groups and their automorphism groups -- 3. Some 2-groups associated with Aut(Z[sub(4)] x Z[sub(4)]) -- 4. Initial reductions -- 5. Elimination of the rank 3 case -- 6. The major reduction -- 7. The non-dihedral case -- 8. The noncyclic case -- 9. The structure of O[sub(2)](M). 10. The structure of S.
9780821899472
Finite groups.
Electronic books.
QA177 -- .G67 1974eb
510/.8 s;512/.2