We show that amalgams of nitely generated torsionfree nilpotent groups of class c along a cyclic subgroup satisfy a polynomial isoperimetric inequality of degree 4c. The distortion of the amalgamated subgroup is bounded above by a polynomial of degree c. We also give an example of a non-cyclic amalgam of nitely generated torsionfree nilpotent groups along an abelian, isolated and normal subgrou...

We prove that the theory of the p-adics Qp, together with a set of explicitly given sorts, admits elimination of imaginaries. Using p-adic integration, we deduce the rationality of certain formal zeta functions arising from definable equivalence relations. As an application, we prove rationality results for zeta functions obtained by counting isomorphism classes of irreducible representations o...

We give the first polynomial-time algorithm for checking whether the Galois group Gal (f) of an input polynomial f(X) ∈ Q[X] is nilpotent: the running time of our algorithm is bounded by a polynomial in the size of the coefficients of f and the degree of f . Additionally, we give a deterministic polynomial-time algorithm that, when given as input a polynomial f(X) ∈ Q[X] with nilpotent Galois g...

A subgroup $H$ is said to be $nc$-supplemented in a group $G$ if there exists a subgroup $Kleq G$ such that $HKlhd G$ and $Hcap K$ is contained in $H_{G}$, the core of $H$ in $G$. We characterize the supersolubility of finite groups $G$ with that every maximal subgroup of the Sylow subgroups is $nc$-supplemented in $G$.

We determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p. We also prove that our result would hold for p-solvable groups, provided a conjecture can be proved about the case of p-nilpotent groups.

Comparative studies on the clinical features and outcomes of triple-negative subgroups to human epidermal growth factor receptor-2 (HER-2) overexpression, and luminal A and B subgroups in lymph node-negative breast cancer patients, are important to correctly evaluate clinical prognosis. A total of 1132 Chinese breast cancer patients were enrolled in a retrospective analysis. We characterized an...

In this paper, we continue our study of the class of diagram groups. Simply speaking, a diagram is a labelled plane graph bounded by a pair of paths (the top path and the bottom path). To multiply two diagrams, one simply identifies the top path of one diagram with the bottom path of the other diagram, and removes pairs of “reducible” cells. Each diagram group is determined by an alphabet X, co...

For a finite group G, subgroup P of G is 2-minimal if B<P, where B=NG(S) for some Sylow 2-subgroup S and B contained in unique maximal P. Here we give detailed explicit description all the subgroups symplectic groups defined over field odd characteristic.

An extraordinary theorem of Gromov, [4], characterizes the finitely generated groups of polynomial growth; a group has polynomial growth iff it is nilpotent by finite. This theorem went a long way from its roots in the class of discrete subgroups of solvable Lie groups. Wolf, [11], proved that a polycyclic group of polynomial growth is nilpotent by finite. This theorem is primarily about linear...

Let G be a rank 2 complete affine or hyperbolic Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let P − 1 = B − t Bw1B be the negative standar...