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Title:  Blocks and virtually irreducible lattices 
Author(s):  Ellers, Harald Erich Herbert 
Doctoral Committee Chair(s):  Dade, Everett C. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  We use R. Knorr's theory of virtually irreducible lattices to study the blocks of a finite group. Let G be a finite group and let p be a rational prime. Let R be a complete discrete valuation ring of characteristic zero with maximal ideal generated by $\pi$ and with p $\varepsilon$ $\pi$R. Let K be the field of fractions of R, and let R = R/$\pi$R. Assume that R is algebraically closed and that K is a splitting field for every subgroup of G. Knorr showed that any indecomposable RGlattice of height zero is virtually irreducible. We use this fact to generalize Brauer's Third Main Theorem on Blocks as follows. Theorem (3.1). Let B be a block of RG, and let M be an indecomposable RGlattice in B of height zero. Suppose that H is a subgroup of G and that b is an admissible block of RH. Then b$\sp{\rm G}$ = B if and only if b contains an indecomposable component of M$\sb{\rm H}$ of height zero. We also prove the following connection between Brauer correspondence of blocks and induction of virtually irreducible lattices. Theorem (5.2). Let H be a subgroup of G and let b be a block of RH. Suppose that there is a virtually irreducible RHlattice U in b such that U$\sp{\rm G}$ = V $\oplus$ W with V virtually irreducible and U $\not\vert$ W$\sb{\rm H}$. Then b$\sp{\rm G}$ is defined and V is in b$\sp{\rm G}$. Most admissible blocks contain a virtually irreducible lattice U as in Theorem (5.2); there is a finite extension S of R such that the following is true. Theorem (5.11). Let H be a subgroup of G and let b be an admissible block of SH with defect group D. If every automorphism of D which preserves conjugacy classes is an inner automorphism, then there is a virtually irreducible SHlattice in b with vertex D such that U$\sp{\rm G}$ = V $\oplus$ W with V virtually irreducible and U $\not\vert$ W$\sb{\rm H}$. We also investigate the question: if B is a block of RG and if there is a block pair (D,b) in G with b$\sp{\rm G}$ = B, is there a virtually irreducible RGlattice in B with vertex D? Theorem (5.11) gives a sufficient condition on D for this question to have an affirmative answer, provided we replace R by a certain finite extension. We give several more conditions of this kind. This is a partial converse to a theorem of Knorr. 
Issue Date:  1989 
Type:  Text 
Language:  English 
URI:  http://hdl.handle.net/2142/19169 
Rights Information:  Copyright 1989 Ellers, Harald Erich Herbert 
Date Available in IDEALS:  20110507 
Identifier in Online Catalog:  AAI9010852 
OCLC Identifier:  (UMI)AAI9010852 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics
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