By Reiner I. (ed.)
Read Online or Download Representation Theory of Finite Groups and Related Topics PDF
Similar group theory books
The illustration idea of finite teams has obvious fast development lately with the improvement of effective algorithms and machine algebra structures. this can be the 1st publication to supply an creation to the standard and modular illustration conception of finite teams with exact emphasis at the computational points of the topic.
This is often the second one of 3 volumes dedicated to straightforward finite p-group concept. just like the 1st quantity, 1000's of significant effects are analyzed and, in lots of instances, simplified. very important issues awarded during this monograph comprise: (a) type of p-groups all of whose cyclic subgroups of composite orders are basic, (b) class of 2-groups with precisely 3 involutions, (c) proofs of Ward's theorem on quaternion-free teams, (d) 2-groups with small centralizers of an involution, (e) class of 2-groups with precisely 4 cyclic subgroups of order 2n > 2, (f) new proofs of Blackburn's theorem on minimum nonmetacyclic teams, (g) class of p-groups all of whose subgroups of index pÂ² are abelian, (h) category of 2-groups all of whose minimum nonabelian subgroups have order eight, (i) p-groups with cyclic subgroups of index pÂ² are labeled.
George Mackey was once a unprecedented mathematician of serious energy and imaginative and prescient. His profound contributions to illustration thought, harmonic research, ergodic conception, and mathematical physics left a wealthy legacy for researchers that keeps this present day. This booklet is predicated on lectures provided at an AMS specified consultation held in January 2007 in New Orleans devoted to his reminiscence.
Additional info for Representation Theory of Finite Groups and Related Topics
0 n with n = proj dim M . Since depth R = 0, the maximal ideal is in Ass R . e. 4, 'n maps Fn isomorphically onto a free direct summand of Fn;1, in contradiction to proj dim M = n. Therefore n = 0, and furthermore depth M = depth R = 0 since M is a free R -module. Let now depth R > 0. Suppose rst that depth M = 0. 9 yields depth M1 = 1 for a rst syzygy M1 of M . Since proj dim M1 = proj dim M ; 1, it is enough to prove the desired formula for M1 . Thus we may assume depth M > 0. Then 2= Ass R and 2= Ass M .
M Proof. For the implication (a) ) (b) it is enough to write every homogeneous element y 2 R as a polynomial in x1 . . xn with coe cients in R0 , and this is very easy by induction on deg y. The rest is evident. 4 holds for graded rings in general. 5. Let R be a graded ring. Then the following are equivalent: (a) every graded ideal of R is nitely generated (b) R is a Noetherian ring (c) R0 is Noetherian, and R is a nitely L generated R0-algebra L (d) R0 is Noetherian, and both S1 = 1i=0 Ri and S2 = 1i=0 R;i are nitely generated R0-algebras.
Let #a denote the multiplication by a on K (f ), and x the left multiplication by x on K (f ). Then #a = df x + x df as is easily veri ed. Thus multiplication by a is null-homotopic on K (f ). Of course #a M and HomR (#a M ) are the multiplications by a on K (f M ) and K (f M ), and the rest of (a) follows immediately. Part (b) is a general fact: if ' is a null-homotopic complex homomorphism, then the map induced by ' on homology is zero. For (c) we choose a = 1, and apply (a) and (b). Let L1 and L2 be R -modules, and f1 : L1 !