Let G be a nontrivial transitive permutation group on a finite set Ω. An element of G is said to be a derangement if it has no fixed points on Ω. From the orbit counting lemma, it follows that G ...contains a derangement, and in fact G contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that G is almost elusive if it contains a unique conjugacy class of derangements of prime order. In this paper we first prove that every quasiprimitive almost elusive group is either almost simple or 2-transitive of affine type. We then classify all the almost elusive groups that are almost simple and primitive with socle an alternating group, a sporadic group, or a rank one group of Lie type.
GroupMath is a Mathematica package which performs several calculations related to semi-simple Lie algebras and the permutation groups, both of which are important in various areas of physics. Having ...in mind the specific needs of theoretical particle physicists, the program computes several basis-independent quantities (such as roots, weights, decomposition of products of representations, branching rules) as well as basis-dependent ones (including explicit representation matrices and Clebsch-Gordon coefficients).
Program Title: GroupMath: A Mathematica package for group theory calculations
CPC Library link to program files:https://doi.org/10.17632/hdkksr6v7t.1
Developer's repository link:renatofonseca.net/groupmath
Licensing provisions: GNU General Public License 3
Programming language: Mathematica
Nature of problem: Computations involving semi-simple Lie algebras are commonplace in several areas of research, such as particle physics. Indeed model builders trying to extend the Standard Model often need to consider new fields which transform as irreducible representations of these algebras. It is therefore convenient to have a computer program which can perform these calculations systematically. Such code might be used directly by the user, or be incorporated in a longer chain of programs. The permutation groups Sm are also important, both by themselves and also as a tool to better describe the properties of the representations of semi-simple Lie algebras.
Solution method: GroupMath performs several computations related to semi-simple Lie algebras which are important in high energy physics: Cartan matrices, roots, weights, Weyl reflections, decomposition of products of representations, subgroups, branching rules, and others. Algorithms available in the literature are used to efficiently perform some of these calculations. The program also computes basis-dependent quantities such as representation matrices and Clebsch-Gordon coefficients. Several elements of the theory of finite permutation groups have also been implemented, including Young diagrams and tableaux, the decomposition of products of representations, branching rules, and explicit representation matrices.
Additional comments including restrictions and unusual features: Most of GroupMath's functions can handle arbitrary representations of any semi-simple Lie algebra and any permutation group. However, memory and time constraints are important when considering large and/or numerous representations. This is particularly true for the functions which perform basis-dependent computations.
Let Please download the PDF to view the mathematical expression be the group of bijections from 0, 1 to itself which are continuous outside a finite set. Let Please download the PDF to view the ...mathematical expression be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of Please download the PDF to view the mathematical expression vanishes. That is, the quotient map Please download the PDF to view the mathematical expression splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, fram Please download the PDF to view the mathematical expression to Z/2Z. Then we use this signature to list normal subgroups of every subgroup G of Please download the PDF to view the mathematical expression which contains Please download the PDF to view the mathematical expression such that G, the projection of G in Please download the PDF to view the mathematical expression, is simple.
We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher–O'Nan–Scott Theorem to all primitive permutation groups with finite ...point stabilizers.
The number of string C-groups of high rank Cameron, Peter J.; Fernandes, Maria Elisa; Leemans, Dimitri
Advances in mathematics (New York. 1965),
September 2024, 2024-09-00, Volume:
453
Journal Article
Peer reviewed
If G is a transitive group of degree n having a string C-group of rank r≥(n+3)/2, then G is necessarily the symmetric group Sn. We prove that if n is large enough, up to isomorphism and duality, the ...number of string C-groups of rank r for Sn (with r≥(n+3)/2) is the same as the number of string C-groups of rank r+1 for Sn+1. This result and the tools used in its proof, in particular the rank and degree extension, imply that if one knows the string C-groups of rank (n+3)/2 for Sn with n odd, one can construct from them all string C-groups of rank (n+3)/2+k for Sn+k for any positive integer k. The classification of the string C-groups of rank r≥(n+3)/2 for Sn is thus reduced to classifying string C-groups of rank r for S2r−3. A consequence of this result is the complete classification of all string C-groups of Sn with rank n−κ for κ∈{1,…,7}, when n≥2κ+3, which extends previously known results. The number of string C-groups of rank n−κ, with n≥2κ+3, of this classification gives the following sequence of integers indexed by κ and starting at κ=1:(1,1,7,9,35,48,135) This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It is available as sequence number A359367.
Let m be a positive integer and let Ω be a finite set. The m-closure of G≤Sym(Ω) is the largest permutation group on Ω having the same orbits as G in its induced action on the Cartesian product Ωm. ...The 1-closure and 2-closure of a solvable permutation group need not be solvable. We prove that the m-closure of a solvable permutation group is always solvable for m≥3.
Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$. We prove that either $k(G)< n/2$ and $k(G)=o(n)$ as ...$n\rightarrow \infty$, or $G$ belongs to explicit families of examples.
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