a subgroup $h$ is said to be $s$-permutable in a group $g$‎, ‎if‎ ‎$hp=ph$ holds for every sylow subgroup $p$ of $g$‎. ‎if there exists a‎ ‎subgroup $b$ of $g$ such that $hb=g$ and $h$ permutes with every‎ ‎sylow subgroup of $b$‎, ‎then $h$ is said to be $ss$-quasinormal in‎ ‎$g$‎. ‎in this paper‎, ‎we say that $h$ is a weakly $ss$-quasinormal‎ ‎subgroup of $g$‎, ‎if there is a normal subgroup ...

A group is small if it has countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions ...

We study the impact of certain identities and probabilistic on structure finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables $G$ group for which map $w_G:G^d\rightarrow G$ has fiber size at least $\rho|G|^d$ some fixed $\rho>0$. show that, words $w$, this implies that normal solvable subgroup index bounded above terms $\rho$. also larger family nonsolvable l...

a result of dixon‎, ‎evans and smith shows that if $g$ is a locally (soluble-by-finite) group whose proper subgroups are (finite rank)-by-abelian‎, ‎then $g$ itself has this property‎, ‎i.e‎. ‎the commutator subgroup of $g$ has finite rank‎. ‎it is proved here that if $g$ is a locally (soluble-by-finite) group whose proper subgroups have minimax commutator subgroup‎, ‎then also the commutator s...

If G is a finite subgroup of the automorphism group of a projective curve X and D is a divisor on X stabilized by G, then we compute a simplified formula for the trace of the natural representation of G on the Riemann-Roch space L(D), under the assumption that L(D) is “rational”, D is nonspecial, and the characteristic is “good”. We discuss the partial formulas that result if L(D) is not rational.

D Laws, E Neville, J Duffy, on behalf of the British Thoracic Society Pleural Disease Group, a subgroup of the British Thoracic Society Standards of Care Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....