UID:
almahu_9949698027202882
Format:
1 online resource (543 p.)
ISBN:
1-281-79800-2
,
9786611798000
,
0-08-087246-8
Series Statement:
North-Holland mathematics studies ; 135
Content:
Let G be a finite group and let F be a field. It is well known that linear representations of G over F can be interpreted as modules over the group algebra FG. Thus the investigation of ring-theoretic structure of the Jacobson radical J(FG) of FG is of fundamental importance. During the last two decades the subject has been pursued by a number of researchers and many interesting results have been obtained. This volume examines these results.The main body of the theory is presented, giving the central ideas, the basic results and the fundamental methods. It is assumed that the reader ha
Note:
Description based upon print version of record.
,
Front Cover; The Jacobson Radical of Group Algebras; Copyright Page; Contents; Preface; CHAPTER 1. RING-THEORETIC BACKGROUND; 1. Notation and terminology; 2. Artinian and noetherian modules; 3. Completely reducible modules; 4. Direct decomposition of rings; 5. Matrix rings; 6. The radical and socle of modules and rings; 7. The Krull-Schmidt theorem; 8. Projective, injective and flat modules; 9. Projective covers; 10. Algebras over fields; CHAPTER 2. GROUP ALGEBRAS AND THEIR MODULES; 1. Group algebras; 2. Central idempotents; 3. The number of irreducible FG-modules; 4. The induced modules
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5. Relative projective and injective modules6. Vertices of FG-modules; CHAPTER 3. THE JACOBSON RADICAL OF GROUP ALGEBRAS: FOUNDATIONS OF THE THEORY; 1. Elementary properties; 2. Direct products; 3. A characterization of elements of J(FG): the general case; 4. Conlon's theorem, Fong's dimension formula and related results; 5. A characterization of elements of J(FG); G is p- solvable; 6. A characterization of elements of J(Z(FG)); 7. Frobenius groups; 8. Upper and lower bounds for dimFJ(Fg); 9. A characterization of dimFJ(Z(FG) ) and its applications; 10. Morita's theorem
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11. An application: criteria for J (FG ) = FG*J(Z(FG))12. Group algebras with radicals of square zero; 13. Group algebras with central radicals; 14. Commutativity of the radical of the principal block; 15. Criteria for the commutativity of J(FG); 16. The radical of blocks and normal subgroups; 17. Group algebras with radicals expressible as principal ideals; CHAPTER 4. GROUP ALGEBRAS OF p-GROUPS OVER FIELDS OF CHARACTERISTIC p; 1. Dimension subgroups in characteristic p 〉 0 and related results; 2. Computation of t(P) for some individual p-groups P
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3. Characterization of groups P of order pa with t(P) = a(p - 1) + 1, t(P) = pa, t(P) = pa-1 + p - 1, t(P) = pa-1 and t(P) = (a+1) (p-1) + 14. Characterizations of p-groups P with t(P) 〈 7; CHAPTER 5. THE JACOBSON RADICAL AND INDUCED MODULES; 1. Annihilators of induced modules; 2. Simple induction and restriction pairs; 3. Applications; 4. p-Radical groups; CHAPTER 6. THE LOEWY LENGTH OF PROJECTIVE MODULES; 1. Preliminary results; 2. The Loewy length of projective covers; 3. The Loewy length of induced modules; 4. Groups of p-length 2; CHAPTER 7. THE NILPOTENCY INDEX
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1. Some results on p-solvable groups2. Upper and lower bounds for t ( G ); 3. Groups G with t ( G ) = a(p - 1) + 1; 4. Computation of t ( G ) with M(p) ? Syl (G); 5. Characterizations of groups G with t(G) =pa - 1 + p - 1; 6 . Groups G with t ( G ) 〈 (a+2) (p - 1) + 1; 7. Characterization of groups G with t ( G ) =pa - 1, p odd; CHAPTER 8. RADICALS OF BLOCKS; 1. A lower bound for t ( B ) in terms of the exponent of 6 ( B); 2. An upper bound for t ( 2 ( B ) ); 3. Defect groups of covering blocks; 4. Regular blocks; 5. The Fong correspondence; 6. The Külshammer's structure theorem
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7. Applications
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English
Additional Edition:
ISBN 0-444-70190-7
Language:
English
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