Generalizing a theorem of P. Hall on finite-by-nilpotent groups

Generalizing a Theorem of P. Hall on Finite-by-nilpotent Groups

Let γi(G) and Zi(G) denote the i-th terms of the lower and upper central series of a group G, respectively. In 1956 P. Hall showed that if γi+1(G) is finite, then the index |G : Z2i(G)| is finite. We prove that the same result holds under the weaker hypothesis that |γi+1(G) : γi+1(G) ∩ Zi(G)| is finite.

NILPOTENT p-LOCAL FINITE GROUPS

In this paper we provide characterizations of p-nilpotency for fusion systems and p-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.

A Note on Absolute Central Automorphisms of Finite $p$-Groups

Let $G$ be a finite group. The automorphism $sigma$ of a group $G$ is said to be an absolute central automorphism, if for all $xin G$, $x^{-1}x^{sigma}in L(G)$, where $L(G)$ be the absolute centre of $G$. In this paper, we study  some properties of absolute central automorphisms of a given finite $p$-group.

On Freiman’s Theorem in Nilpotent Groups

We generalize a result of Tao which extends Freiman’s theorem to the Heisenberg group. We extend it to simply connected nilpotent Lie groups of arbitrary step.

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup