AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES

Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automorphisms simply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications

Transitivity in finite general linear groups

It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme of $\operatorname{GL}(n,q)$. In particular we generalise a theorem of Perin on subgroups of $\operatorname{GL}(n,q)$ acting transitively on $t$-dimensional subspaces. We survey transitive subgroups of $\operatorname{GL}(n,q)$, showing that there is no subgroup of $\operatorname{GL}(n,q)$ with $1<t<n$ acting transitively on $t$-dimensional subspaces unless it contains $\operatorname{SL}(n,q)$ or is one of two exceptional groups. On the other hand, for all fixed $t$, we show that there exist nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on linearly independent $t$-tuples of $\mathbb{F}_q^n$, which also shows the existence of nontrivial subsets of $\operatorname{GL}(n,q)$ that are transitive on more general flag-like structures. We establish connections with orthogonal polynomials, namely the Al-Salam-Carlitz polynomials, and generalise a result by Rudvalis and Shinoda on the distribution of the number of fixed points of the elements in $\operatorname{GL}(n,q)$. Many of our results can be interpreted as $q$-analogs of corresponding results for the symmetric group.Comment: 28 page

Between primitive and 2-transitive : synchronization and its friends

The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.PostprintPeer reviewe

Design of advanced primitives for secure multiparty computation : special shuffles and integer comparison

In modern cryptography, the problem of secure multiparty computation is about the cooperation between mutually distrusting parties computing a given function. Each party holds some private information that should remain secret as much as possible throughout the computation. A large body of research initiated in the early 1980's has shown that any computable function can be evaluated using secure multiparty computation. Though these feasibility results are general, their applicability in practical situations is rather unsatisfactory. This thesis concerns the study of two particular cryptographic primitives with focus on efficiency. The first primitive studied is a generalization of verifiable shuffles of homomorphic encryptions, where the shuffler is only allowed to apply a permutation from a restricted set of permutations. In this thesis, we consider shuffles using permutations from a k-fragile set, meaning that any k input-output correspondences uniquely identify a permutation within the set. We provide verifiable shuffles restricted to the set of all rotations (1-fragile), affine transformations (2-fragile), and Möbius transformations (3-fragile). Applications of these special shuffles include fragile mixing, electronic elections, secure function evaluation using scrambled circuits, and secure integer comparison. Two approaches for verifiable rotations are presented. On the one hand, we use properties of the Discrete Fourier Transform (DFT) to express in a compact way that a rotation is applied in a shuffle. The solution is efficient, but imposes some mild restrictions on the parameters to allow DFT to work. On the other hand, we present a general solution that does not impose any parameter constraint and works on any homomorphic cryptosystem. These protocols for rotations are used to build efficient shuffling protocols for affine and Möbius transformations. The second primitive is secure integer comparison. In a general scenario, parties are given homomorphic encryptions of the bits of two integers and, after running a protocol, an encryption of a bit is produced, telling the result of the greater-than comparison of the two integers. This is a useful building block for higher-level protocols such as electronic voting, biometrics authentication or electronic auctions. A study of the relationship of other problems to integer comparison is given as well. We present two types of solutions for integer comparison. Firstly, we consider an arithmetic circuit yielding secure protocols within the framework for multiparty computation based on threshold homomorphic cryptosystems. Our circuit achieves a good balance between round and computational complexities, when compared to the similar solutions in the literature. The second type of solutions uses a intricate approach where different building blocks are used. A full analysis is made for the two-party case where efficiency of the resulting protocols compares favorably to other solutions and approaches

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Construction and Simplicity of the Large Mathieu Groups

In this thesis, we describe the construction of the Mathieu group M24 given by Ernst Witt in 1938, a construction whose geometry was examined by Jacques Tits in 1964. This construction is achieved by extending the projective semilinear group PΓL3(F4) and its action on the projective plane P²(F4). P²(F4) is the projective plane over the field of 4 elements, with 21 points and 21 lines, and PΓL3(F4) is the largest group sending lines to lines in P²(F4).This plane has 168 6-point subsets, hexads, with the property that no 3 points of a hexad are collinear. Under the action of the subgroup PSL3(F4), the hexads in P²(F4) break into 3 orbits of equal size. These orbits are preserved and permuted by PΓL3(F4), and can be viewed as 3 points, which, when added to the 21 points of P²(F4), yield a set X of 24 points. Using lines and hexads in P²(F4), we define certain 8-point subsets of X, view them as vectors in F224 and define the subspace they span as the Golay 24-code. We then define M24 as the automorphism group of the Golay 24-code and show that it acts 5-transitively on X, establishing its simplicity. We calculate the order of M24 and the order of two simple subgroups, M23and M22, the other large Mathieu groups

Why must we work in the phase space?

We are going to prove that the phase-space description is fundamental both in the classical and quantum physics. It is shown that many problems in statistical mechanics, quantum mechanics, quasi-classical theory and in the theory of integrable systems may be well-formulated only in the phase-space language.Comment: 130 page