A finite non-abelian group G is called metahamiltonian if every subgroup of either abelian or normal in G. If non-nilpotent, then the structure has been determined. nilpotent, determined by its Sylow subgroups. However, classification p-groups an unsolved problem. In this paper, are completely classified up to isomorphism.

We describe an algorithm to compute a composition tree for a matrix group defined over a finite field, and show how to use the associated structure to carry out computations with such groups; these include finding composition and chief series, the soluble radical, and Sylow subgroups.

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer D 0; 1 mod 4 and the factorization of D, computes the structure of the 2-Sylow subgroup of the class group of the quadratic order of discriminant D in random polynomial time in log jDj.

We fix a monic polynomial f(x) ∈ Fq[x] over a finite field and consider the Artin-Schreier-Witt tower defined by f(x); this is a tower of curves · · · → Cm → Cm−1 → · · · → C0 = A, with total Galois group Zp. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tow...

This paper deals with the determination of the automorphism group of the metacyclic p-groups, P (p,m), given by the presentation P (p,m) = 〈x, y|xpm = 1, y = 1, yxy−1 = xp+1〉 (1) where p is an odd prime number and m > 1. We will show that Aut(P ) has a unique Sylow p-subgroup, Sp, and that in fact

We prove that a knowledge of the character degrees of a finite group G and of their multiplicities determines whether G has a Sylow p-subgroup as a direct factor. An analogous result based on a knowledge of the conjugacy class sizes was known. We prove variations of both results and discuss their similarities.

We complete the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups.

Suppose that  is a finite group. Then the set of all prime divisors of  is denoted by  and the set of element orders of  is denoted by . Suppose that . Then the number of elements of order  in  is denoted by  and the sizes of the set of elements with the same order is denoted by ; that is, . In this paper, we prove that if  is a group such that , where , then . Here  denotes the family of Suzuk...

Let G be a permutation group of set ? and k positive integer. The k-closure is the greatest (w.r.t. inclusion) subgroup G(k) in Sym(?) which has same orbits as under componentwise action on ?k. It proved that finite nilpotent coincides with direct product k-closures all its Sylow subgroups.