Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...

Let A be a basic, connected nite-dimensional algebra over an algebraically closed eld with nite representation type, (A) be the maximum of the set consisting of numbers of indecomposable summands in the middle term of Auslander-Reiten sequences in modA. In this paper I show that if (A)=1, then ? A contains oriented cycles if and only if ? A contains DTr-periodic modules. When (A) 2 I give count...

Given a regular action of a nite group G on a set V , we ask (and answer) the question of the existence of an incidence structure I = (V; B) on the set V whose full automorphism group Aut(I) is the group G in its regular action. Additional conditions on I also allow us to reene the original problem to the class of hypergraphs. Using results on graphical and digraphical regular representations (...

We introduce generalized nite automata as a tool for speciication of bi-level images. We describe an inference algorithm for generalized nite automata and a lossy compression system for bi-level and simple color images based on this algorithm and vector quantization.

We give a new and simple proof of a key lemma on the rank of a matrix in Leung's paper [3], where he showed there exists a family of nondeterministic nite automata fAng, with n states and exponential ambiguity, such that any smallest equivalent polynomially ambiguous nite automaton has 2 1 states. Keyword: nondeterministic automata, ambiguity, automata theory.

Let G be a nite group acting as algebra automorphisms on A := FX 1 ; : : : ; Xn]. If F is a eld of characteristic zero, then, due to classical results of Emmy Noether one knows that the invariant ring A G can be generated in degrees less or equal to jGj. If F an arbitrary commutative ring (e.g. a nite eld), the situation is much less satisfying. In this paper we give an outline on known results...

The purpose of this note is to prove the following: Theorem 1.1 Let G be a nite group. Then there is a rational homology S 3 on which G acts freely. That any nite group acts freely on some closed 3-manifold is easy to arrange: There are many examples of closed 3-manifolds whose fundamental groups surject a free group of rank two (for example, by taking a connected sum of S 1 S 2 's) and by pass...

Chiral orbifold models are deened as gauge eld theories with a nite gauge group ?. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative deenite integral invariant bilinear form on its Lie algebra. Any nite group ? of inner automorphisms or A (in particular, any nite subgroup of G) gives rise to a gauge theory with a chiral subalgebra A ? A o...

We study extensions of p-local nite groups where the kernel is a pgroup. In particular, we construct examples of saturated fusion systems F which do not come from nite groups, but which have normal p-subgroups A C F such that F=A is the fusion system of a nite group. One of the tools used to do this is the concept of a \transporter system", which is modelled on the transporter category of a nit...

Let G be a simple primitive subgroup of Sn, speciied in terms of a set of generating permutations. If jGj n 5 , eecient algorithms are presented that nd \the most natural permutation representation" of G. For example, if G is a classical group then we nd a suitable projective space underlying G. A number of related questions are considered. Our notion of \eeciency" takes into account many exist...