On the existence of normal subgroups of prime index

In this article, we characterize finite groups having normal subgroups of a given prime index. Precisely, we prove that if p is a prime divisor of a finite group G, then G has no normal subgroup of index p if and only if G=G’Gp, where Gp is the subgroup of G generated by all elements of the form gp for any g in G and G’ is the derived subgroup of G. We also extend a characterization of finite groups with no subgroups of index 2 by J.B. Nganou to infinite groups. We display an example to show that for a prime index greater than 2 the characterization does not hold

Detecting large groups

Let G be a finitely presented group, and let p be a prime. Then G is 'large' (respectively, 'p-large') if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a variety of new methods for detecting whether G is large or p-large. These relate to the group's profinite and pro-p completions, to its first L2-Betti number and to the existence of certain finite index subgroups with 'rapid descent'. The paper draws on new topological and geometric techniques, together with a result on error-correcting codes.Comment: 31 pages, 2 figure

THE SYLOW THEOREM AND ITS CONSEQUENCES

The aim of the paper is to present some problems and also some partial results mainly on groups and converse of langrage’s theorem with the help of Sylow theorems.in this paper we find different sylow sub-groups and deduce the normalizer of sylow subgroups. Key words: Finite group, the number of subgroups of prime power, index in a group  of the normalizer of any Sylow subgroups

The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence" subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta<1 is the Hausdorff dimension of the limit set of Gamma. If delta>149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.Comment: 33 pages, 1 figure, minor corrections. To appear, Duke Math