A group of order n is said to be R-sequenceable if the nonidentity elements of the group can be listed in a sequence a,, a,, . . . , a,_ 1 such that the quotients a; ‘az, a; ’ a3, . , a;?,a,_ r, a.-_‘,a, are distinct. An abelian group is R*-sequenceable if it has an R-sequencing a,, a,, . . , a,, _ 1 such that a,_ ra, + r = ai for some i (subscripts are read modulo n 1). Friedlander, Gordon and...

The third Conway group Co3 is one of the twenty-six sporadic finite simple groups. The cohomology of its Sylow 2-subgroup S is computed, an important step in calculating the mod 2 cohomology of Co3. The spectral sequence for the central extension of S is described and collapses at the sixth page. Generators are described in terms of the Evens norm or transfers from subgroups. The central quotie...

This report presents a formalisation of Sylow’s theorems done in Coq. The formalisation has been done in a couple of weeks on top of Georges Gonthier’s ssreflect [2]. There were two ideas behind formalising Sylow’s theorems. The first one was to get familiar with Georges way of doing proofs. The second one was to contribute to the collective effort to formalise a large subset of group theory in...

On the Automorphism Tower of Free Nilpotent Groups Martin Dimitrov Kassabov 2003 In this thesis I study the automorphism tower of free nilpo-tent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. Th...

A finite group $G$ is called rational if all its irreducible complex characters are rational valued. In this paper we discuss about rational groups with Sylow 2-subgroups of nilpotency class at most 2 by imposing the solvability and nonsolvability assumption on $G$ and also via nilpotency and nonnilpotency assumption of $G$.

Let X be a smooth projective connected curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Let G be a finite group, P a Sylow p-subgroup of G and NG(P ) its normalizer in G. We show that if there exists an étale Galois cover Y → X with group NG(P ), then G is the Galois group wan étale Galois cover Y → X , where the genus of X depends on the order of G, th...

The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year with the use of the classification of finite simple groups (CFSG). A series of interesting questions arise naturally. First of all, it is not clear whether it is possible to avoid C...