Browsing by Author "Hulpke, Alexander, advisor"
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Item Open Access Arithmetic in group extensions using a partial automation(Colorado State University. Libraries, 2010) Ziliak, Ellen, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Betten, Anton, committee member; McConnell, Ross, committee memberThe purpose of this paper is to describe the structure of an extension group G which has a normal subgroup K and a quotient group Q = G/K . To describe the structure of G concretely, we want to be able to do arithmetic in G based on the arithmetic done in both the normal subgroup K and the quotient group Q. We will begin by looking at the 2-cohomology group which is the standard approach for working on this problem. This will lead us to questions concerning storage which we would like to reduce. Therefore we will consider the case where our groups arc finitely presented and see how storage may be reduced. During this reduction we will see that it will be necessary to be able to rewrite words in a free group as a product of generators of the normal subgroup K. We begin by looking at current approaches to this problem, which requires computing an (augmented) coset table. If we will let Q be a finite group for which we also have a presentation < S\R >, (i.c. Q = F/N with F —< S > and N the normal closure of R in F). We assume that Q does not have a confluent rewriting system. We want to rewrite a word in S, representing the identity in Q as a product of conjugates in R. Such rewriting can be done using an (augmented) coset table for N in F which can be visualized in a graph by a coset automaton, also called the full Cayley Graph. Tracing in the graph through words in F will allow us to rewrite these words as a product of generators of N. The difficulty that arises in this approach lies in storing and constructing the augmented coset table. Instead we will construct an object called a partial automaton which is a subgraph of the coset automaton. The partial automaton will have the property that it contains a loop for every relator in R. We will first show that this graph can be used to reconstruct the coset automaton, which means it contains the same information as the coset automaton even though it is much smaller. Our next step will be to use the partial automaton to rewrite words in N as a product of conjugates in R. Since the partial automaton is much smaller than the coset automaton, and it does not contain doubly labeled edges as an augmented coset automaton would it require substantially less memory to store. A word in N is represented by a loop in the coset automaton, therefore if we wish to rewrite this word as a product of conjugates of relators, we essentially want to describe this larger loop as a product of smaller loops. Where we will restrict our smaller loops to be loops in the partial automaton. To do this rewriting we place the partial automaton locally at different states in the coset automaton until we cover the entire loop. By placing the partial automaton at different states in the graph we will then the conjugate of relators. Unfortunately we cannot just place the partial automaton arbitrarily at different states, because we would have many different choices of the conjugates of relators we could choose. Instead we must use one further tool, which is the fact that our normal subgroup N is itself a free group. Therefore N has a free generating set, where the generators of N are conjugates of relators. With this generating set we can rewrite words in N uniquely as a product of the generators. We will therefore, use the partial automaton to compute the generators of the free generating set for N and then use these generators to rewrite our word in N as a product of conjugate of relators. By using the partial automaton to do this rewriting we can quickly do rewriting in much larger examples. This algorithm has been implemented in GAP and to suggest the improvement we rewrote several words in the group PSp^ which is a group of order 1,451, 520. The partial automaton had 145 states and after some initial set up which will be described in the paper the run time for this rewriting took less than a half a second per word.Item Open Access Automorphism towers of general linear groups(Colorado State University. Libraries, 2008) Jónsdóttir, Margrét Sóley, author; Hulpke, Alexander, advisorLet G0, be a group, G 1 be the automorphism group of G0, G2 the automorphism group of G1 etc. The sequence of these groups together with the natural homomorphisms πi,i+1 : Gi → Gi+1, which take each element to the inner automorphism it induces, is called the automorphism tower of G 0. If πi,i+1 is an isomorphism for some i then the automorphism tower of G is said to terminate. For a given group it is in general not easy to say whether its automorphism tower terminates. Wielandt showed in 1939 that if G is finite with a trivial center then the automorphism tower of G will terminate in a finite number of steps. Since then, some sporadic examples of automorphism towers of finite groups have been described but no general results have been proven. In this thesis we study automorphism towers of finite groups with a non-trivial center. We look at the two extremes: (1) Groups which are center-rich. (2) Groups which have a small but non-trivial center. We show that when looking for an infinite family of groups with terminating automorphism towers the first case is unfeasible. We then turn our attention to the latter case, specifically general linear groups of dimension at least two. In odd characteristic GL(2, q) is not a split extension of the center. The first thing we do is to calculate the automorphism group of GL(2, q) for odd prime powers q. We provide explicit generators and describe the structure of Aut(GL(2, q)) in terms of well-known groups. In this case, the first automorphism group in the tower is a subdirect product of two characteristic factors. This structure is propagated through the tower and we use it to reduce the problem to studying subgroups of automorphism groups of smaller groups. We then use this structure to compute examples of automorphism towers of GL(2, q).Item Open Access Determining synchronization of certain classes of primitive groups of affine type(Colorado State University. Libraries, 2022) Story, Dustin, author; Hulpke, Alexander, advisor; Adams, Henry, committee member; Buchanan, Norm, committee member; Gillespie, Maria, committee memberThe class of permutation groups includes 2-homogeneous groups, synchronizing groups, and primitive groups. Moreover, 2-homogeneous implies synchronizing, and synchronizing in turn implies primitivity. A complete classification of synchronizing groups remains an open problem. Our search takes place amongst the primitive groups, looking for examples of synchronizing and non-synchronizing. Using a case distinction from Aschbacher classes, our main results are constructive proofs showing that three classes of primitive affine groups are nonsynchronizing.Item Open Access Group action on neighborhood complexes of Cayley graphs(Colorado State University. Libraries, 2014) Hughes, Justin, author; Hulpke, Alexander, advisor; Peterson, Chris, advisor; Berger, Bruce, committee member; Cavalieri, Renzo, committee member; Wilson, James, committee memberGiven G a group generated by S ≐ {g1, …, gn}, one can construct the Cayley Graph Cayley (G,S). Given a distance set D ⊂ Z≥0 and Cayley (G,S) one can construct a D-neighborhood complex. This neighborhood complex is a simplicial complex to which we can associate a chain complex. The group G acts on this chain complex and this leads to an action on the homology of the chain complex. These group actions decompose into several representations of G. This thesis uses tools from group theory, representation theory, homo-logical algebra, and topology to further our understanding of the interplay between generated groups (i.e. a group together with a set of generators), corresponding representations on their associated D-neighborhood complexes, and the homology of the D-neighborhood complexes.Item Open Access Independence complexes of finite groups(Colorado State University. Libraries, 2021) Pinckney, Casey M., author; Hulpke, Alexander, advisor; Peterson, Chris, advisor; Adams, Henry, committee member; Neilson, James, committee memberUnderstanding generating sets for finite groups has been explored previously via the generating graph of a group, where vertices are group elements and edges are given by pairs of group elements that generate the group. We generalize this idea by considering minimal generating sets (with respect to inclusion) for subgroups of finite groups. These form a simplicial complex, which we call the independence complex. The vertices of the independence complex are nonidentity group elements and the faces of size k correspond to minimal generating sets of size k. We give a complete characterization via constructive algorithms, together with enumeration results, for the independence complexes of cyclic groups whose order is a squarefree product of primes, finite abelian groups whose order is a product of powers of distinct primes, and the nonabelian class of semidirect products Cp1p3…p2n-1 rtimes Cp2p4…p2n where p1,p2,…,p2n are distinct primes with p2i-1 > p2i for all 1 ≤ i ≤ n. In the latter case, we introduce a tool called a combinatorial diagram, which is a multipartite simplicial complex under certain numerical and minimal covering conditions. Combinatorial diagrams seem to be an interesting area of study on their own. We also include GAP and Polymake code which generates the facets of any (small enough) finite group, as well as visualize the independence complexes in small dimensions.Item Open Access Modular group and modular forms(Colorado State University. Libraries, 2010) Schmidt, Eric, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Cavalieri, Renzo, committee member; Harton, John, committee memberWe prove some results about the structure of SL2(Z) and related groups. We define modular forms for this group and develop the basic theory. We then use the theory of lattices to construct examples of modular forms.Item Open Access The conjugacy extension problem(Colorado State University. Libraries, 2021) Afandi, Rebecca, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Pries, Rachel, committee member; Rajopadhye, Sanjay, committee memberIn this dissertation, we consider R-conjugacy of integral matrices for various commutative rings R. An existence theorem of Guralnick states that integral matrices which are Zp-conjugate for every prime p are conjugate over some algebraic extension of Z. We refer to the problem of determining this algebraic extension as the conjugacy extension problem. We will describe our contributions to solving this problem. We discuss how a correspondence between Z-conjugacy classes of matrices and certain fractional ideal classes can be extended to the context of R-conjugacy for R an integral domain. In the case of integral matrices with a fixed irreducible characteristic polynomial, this theory allows us to implement an algorithm which tests for conjugacy of these matrices over the ring of integers of a specified number field. We also describe how class fields can be used to solve the conjugacy extension problem in some examples.Item Open Access The D-neighborhood complex of a graph(Colorado State University. Libraries, 2014) Previte, Corrine, author; Peterson, Chris, advisor; Hulpke, Alexander, advisor; Bates, Dan, committee member; Gelfand, Martin, committee memberThe Neighborhood complex of a graph, G, is an abstract simplicial complex formed by the subsets of the neighborhoods of all vertices in G. The construction of this simplicial complex can be generalized to use any subset of graph distances as a means to form the simplices in the associated simplicial complex. Consider a simple graph G with diameter d. Let D be a subset of {0,1,..., d}. For each vertex, u, the D-neighborhood is the simplex consisting of all vertices whose graph distance from u lies in D. The D-neighborhood complex of G, denoted DN(G,D), is the simplicial complex generated by the D-neighborhoods of vertices in G. We relate properties of the graph G with the homology of the chain complex associated to DN(G,D).Item Open Access The group extensions problem and its resolution in cohomology for the case of an elementary abelian normal sub-group(Colorado State University. Libraries, 2018) Adams, Zachary W., author; Hulpke, Alexander, advisor; Patel, Amit, committee member; Bohm, Wim, committee memberThe Jordan-Hölder theorem gives a way to deconstruct a group into smaller groups, The converse problem is the construction of group extensions, that is to construct a group G from two groups Q and K where K ≤ G and G/K ≅ Q. Extension theory allows us to construct groups from smaller order groups. The extension problem then is to construct all extensions G, up to suitable equivalence, for given groups K and Q. This talk will explore the extension problem by first constructing extensions as cartesian products and examining the connections to group cohomology.Item Open Access The Möbius number of the symmetric group(Colorado State University. Libraries, 2012) Monks, Kenneth M., author; Hulpke, Alexander, advisor; Penttila, Tim, committee member; Achter, Jeff, committee member; Toki, Walter, committee memberThe Möbius number of a finite group is its most important nontrivial combinatorial invariant. In this paper, we compute the Möbius numbers of many partially-ordered sets, including the odd-partition posets and the subgroup lattices of many infinite families of groups. This is done with an eye towards computing the Möbius number of the symmetric group on 18 points.