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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 28 Mar 2024 10:42:29 -0700Homotopy and homology of p-subgroup complexes
https://resolver.caltech.edu/CaltechETD:etd-06032004-143153
Year: 1994
DOI: 10.7907/GAWP-0T18
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...].
Abstract is included in .pdf document.
In this thesis we analyzed the simple connectivity of the Quillen complex at [...] for the classical groups of Lie type. In light of the Solomon-Tits theorem, we focused on the case where [...] is not the characteristic prime. Given (p,q) = 1. let dp(q) be the order of [...] in [...]. In this thesis we proved the following result:
Main Theorem. When (p,q) = 1 we have the following results about the simple connectivity of the Quillen complex at p, Ap(G), for the classical groups of Lie type:
1. If G = GLn(q), dp(q) > 2 and mp(G) > 2, then Ap[...](G) is simply connected.
2. If G = [...], then:
(a) Ap(G) is Cohen-Macaulay of dimension n - 1 if dp(q) = 1.
(b) If nip(G) > 2 and dp(q) is odd, then Ap(G) is simply connected.
3. If G = [...], then:
(a) Ap(G) is Cohen-IVlacaulay of dimension n - 1 if [...] and dp(q) = 1.
(b) If mp(G) > 2 and dp(q) is odd, then ,Ap(G) is simply connected.
(c) If n [...] 3, q [...] 5 is odd, and dp(q) = 2, then Ap(G)(> Z) is simply connected, where Z is the central subgroup of G of order p.
In the course of analyzing the [...]-subgroup complexes we developed new tools for studying relations between various simplicial complexes and generated results about the join of complexes and the [...]-subgroup complexes of products of groups. For example we proved:
Theorem A. Let [...] be a map of posets satisfying:
(1) [...] is strict; that is,[...]
(2)[...]
(3) [...]connected for all [...] with [...].
Then Y n-connected implies X is n-connected.
Theorem A provides us with a tool for studying [...] in terms of [...]. For example, we used this method to prove:
Theorem 8.6. Let G = [...] where [...] is solvable and S is a p-group of
symplectic type. Then [...]spherical.
In this thesis we also generated a library of results about geometric complexes which do not arise as [...]-subgroup complexes. This library includes, but is not restricted to, the following:
(l.) the poset of proper nondegenerate subspaces of a 2[...]-dimensional symplectic space -ordered by inclusion - is Cohen-Macaulay of dimension n-2.
(2) If q is an odd prime power anal n [...] (with n [...] 5 if q = 3), then the poset of proper nondegenerate subspaces of an n-dimensional unitary space over Fq2 is simply connected.https://resolver.caltech.edu/CaltechETD:etd-06032004-143153Bounds of fixed point ratios of permutation representations of GL_n(q) and groups of genus zero
https://resolver.caltech.edu/CaltechTHESIS:04112011-134618813
Year: 1991
DOI: 10.7907/a3e6-tj54
If G is a transitive subgroup of the symmetric group Sym (Ω), where Ω is a finite set of order m; and G satisfies the following conditions: G=<S>, S={g_1,…,g_r] ⊆ G^#, g_1…g_r=1, and r∑i=1 c(g_i)=(r-2)m+2, where c(g_i) is the number of cycles of g_1 on Ω, then G is called a group of genus zero. These conditions correspond to the existence of an m-sheeted branched covering of the Riemann surface of genus zero with r branch points. The fixed point ratio of an element g in G is defined as f(g)/|Ω|, where f(g) is the number of fixed points of g on Ω. In this thesis we assume that G satisfies L_n(q) ≤G≤PGL_n(q) and G is represented primitively on Ω. The primitive permutation representations of G are determined by the maximal subgroups of G. The bounds are expressed as rational functions which depend on n, q, the rational canonical forms of the elements, and the maximal subgroups. Then those bounds are used to prove the following: Theorem: If G is a group of genus zero, then one of the following holds: (a) q=2 and n≤32, (b) q=3 and n≤12, (c) q=4 and n≤11, (d) 5≤q≤13 and n≤8, (e) 16≤q≤83 and n≤4, (f) 89≤q≤343 and n=2.
Thus for those G satisfying L_n(q) ≤G≤PGLn(q), this theorem confirms the J. Thompson’s conjecture which states that except for Z_p, A_k with k≥5, there are only finitely many finite simple groups which are composition factors of groups of genus zero.
https://resolver.caltech.edu/CaltechTHESIS:04112011-134618813The maximal subgroups of the Chevalley groups F4(F) where F is a finite or algebraically closed field of characteristic not equal to 2,3
https://resolver.caltech.edu/CaltechETD:etd-06132007-094324
Year: 1990
DOI: 10.7907/D2GB-VK65
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
We find the conjugacy classes of maximal subgroups of the almost simple groups of type F4(F), where F is a finite or algebraically closed field of characteristic not equal to 2,3. To do this we study F4(F) via its representation as the automorphism group of the 27-dimensional exceptional central simple Jordan Algebra J defined over F. A Jordan Algebra over a field of characteristic not equal to 2 is a nonassociative algebra over a field F satisfying xy = yx and [...] = [...] for all its elements x and y. We can represent Aut(F4(F)) on J as the group of semilinear invertible maps preserving the multiplication. Let G = F4(F) and [...]. We have defined a certain subset of proper nontrivial subalgebras as good. The principal results are as follows: SUBALGEBRA THEOREM: Let F be a finite or algebraically closed field of characteristic not equal to 2,3. Let H be a subgroup of [...] and suppose that H stabilizes a subalgebra. Then H stabilizes a good subalgebra. The conjugacy classes and normalizers of good subalgebras are also given.
STRUCTURE THEOREM: Let H be a subgroup of [...] such that [...] is closed but not almost simple. Then H stabilizes a proper nontrivial subalgebra or H is contained in a conjugate of [...]. The action of [...] on J is described and it is shown that [...] is unique up to conjugacy in G.
THEOREM : If L is a closed simple nonabelian subgroup of G, then [...] is maximal in [...] only if L is one of the following: [...]. For each member [...] we identify those representations [...] which could give rise to a maximal subgroup of G and show the existence of [...] in G. Up to few exceptions we also determine the number of G conjugacy classes for each equivalence class [...].https://resolver.caltech.edu/CaltechETD:etd-06132007-094324Dade's Ordinary Conjecture for the Finite Unitary Groups in the Defining Characteristic
https://resolver.caltech.edu/CaltechTHESIS:11212019-153036477
Year: 1999
DOI: 10.7907/xhe3-q841
<p>There has been rising interest in the study of Dade's conjectures, which not only generalize Alperin's weight conjecture, but unify some other major conjectures in (modular) representation theory, such as Brauer's height conjecture in abelian blocks and McKay’s conjecture. In this thesis we verify Dade's ordinary conjecture for the finite unitary groups in the defining characteristic. Dade's conjectures involve proving the vanishing of the alternating sum of certain <i>G</i>-stable function over the <i>p</i>-group complex of a finite group <i>G</i>. We develop some machinery to treat alternating sums which we hope will serve as part of a general approach to such problems. This includes extending some of the existing techniques in a functorial way. We also show how to make use of the topological properties of <i>p</i>-group complexes to reduce the alternating sums. While this work is mainly intended for the unitary groups, it should also apply to other groups of Lie type, and part of the work can be generalized to treat a much wider class of groups. Among other things, we also obtain a formula which expresses the McKay's numbers of the finite unitary groups in term s of partitions of integers.</p>https://resolver.caltech.edu/CaltechTHESIS:11212019-153036477On the Mumford-Tate Conjecture for Abelian Varieties with Reduction Conditions
https://resolver.caltech.edu/CaltechTHESIS:05082013-153012294
Year: 1994
DOI: 10.7907/g2mp-jn54
<p>In this thesis we study Galois representations corresponding to abelian varieties
with certain reduction conditions. We show that these conditions force the image
of the representations to be "big," so that the Mumford-Tate conjecture (:= MT)
holds. We also prove that the set of abelian varieties satisfying these conditions is
dense in a corresponding moduli space. </p>
<p>The main results of the thesis are the following two theorems. </p>
<p>Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k :
imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad
reduction at some prime ϕ, with the dimension of the toric part of the reduction
equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds. </p>
<p>Theorem B: Let M be the moduli space of abelian varieties with fixed polarization,
level structure and a k-action. It is defined over a number field F. The subset
of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable
reduction at a large enough prime ϕ of F is dense in M(C) in the complex
topology. In particular, the set of simple abelian varieties having bad reductions
with fixed dimension of the toric parts is dense. </p>
<p>Besides this we also established the following results: </p>
<p> (1) MT holds for some other classes of abelian varieties with similar reduction
conditions. For example, if A is an abelian variety with End° (A) = Q and
the dimension of the toric part of its reduction is prime to dim( A), then MT
holds. </p>
<p> (2) MT holds for Ribet-type abelian varieties. </p>
<p> (3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds. </p>
<p> (4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some
4-folds of type I. </p>
<p> (5) For some abelian varieties either MT or the Hodge conjecture holds. </p>
https://resolver.caltech.edu/CaltechTHESIS:05082013-153012294