Let n be a positive integer or infinity (denote ∞). We denote by W ∗(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X0 ⊆ X, with 2 ≤ |X0| ≤ n + 1 and a function f : {0, 1, 2, . . . , k} −→ X0, with f(0) 6= f(1) and non-zero integers t0, t1, . . . , tk such that [x0 0 , x t1 1 , . . . , x tk k ] = 1, where xi := f(...

Let M be an irreducible, orientable, closed 3-manifold with fundamental group G. We show that if the pro-p completion Ĝp of G is infinite then G is either soluble-by-finite or contains a free subgroup of rank 2.

We describe a significantly improved algorithm for computing the conjugacy classes of a finite permutation group with trivial soluble radical. We rely on existing methods for groups that are almost simple, and we are concerned here only with the reduction of the general case to the almost simple case. Dedicated to Charles Leedham-Green on the occasion of his 65th birthday

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.

Define a sequence (sn) of two-variable words in variables x, y as follows: s0(x, y) = x, sn+1(x, y) = [sn(x, y)−y, sn(x, y)] for n > 0. It is shown that a finite group G is soluble if and only if sn is a law of G for all but finitely many values of n.

Let σ={σi:i∈I} be a partition of the set all prime numbers. A subgroup H finite group G is said to σ-subnormal in if can joined by chain subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 normal Hj or Hj/CoreHj(Hj−1) σi-group some i∈I. B soluble normalising Nσ-residual non-σ-subnormal G, where Nσ saturated formation σ-nilpotent groups. We show that normalises does not have section σ-residu...

Let $G$ be a finite non-abelian $p$-group and $L(G)$ denotes the absolute center of $G$. Also, let $Aut^{L}(G)$ and $Aut_c(G)$ denote the group of all absolute central and the class preserving automorphisms of $G$, respectively. In this paper, we give a necessary and sufficient condition for $G$ such that $Aut_c(G)=Aut^{L}(G)$. We also characterize all finite non-abelian $p$-groups of order $p^...

Let $G$ be a finite primitive permutation group on set $\Omega$ with point stabiliser $H$. Recall that subset of is base for if its pointwise trivial. We define the size $G$, denoted $b(G,H)$, to minimal $G$. Determining fundamental problem in theory, long history stretching back 19th century. Here one our main motivations theorem Seress from 1996, which states $b(G,H) \leqslant 4$ soluble. In ...

G. Navarro raised the question whether the ordinary character table X(G) of a finite group G determines the Sylow numbers of G. In this note we show that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are determined by their ordinary character table. If G and H are finite groups with i...