We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |XXX|/|X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences s...

Dual numbers, split-quaternions, split-octonions, and other number systems with nilpotent spaces have received sporadic yet persistent interest, beginning from their roots in the 19th century, to more recent attention in connection with supersymmetry in physics. In this paper, a number system in the 2D plane is investigated, where the squares of its basis elements p and q each map into the coor...

Abstract Residual torsion-free nilpotence has proved to be an important property for knot groups with applications bi-orderability and ribbon concordance. Mayland proposed a strategy show that two-bridge group commutator subgroup which is union of ascending chain para-free groups. This paper proves Mayland’s assertion expands the result subgroups link correspond kernels maps $\mathbb{Z}$ . We c...

Suppose that G is a finite group and k field of characteristic $$p>0$$ . We consider the complete cohomology ring $${\mathcal {E}}_M^* = \sum _{n \in {\mathbb Z}} \widehat{{\text {Ext}}}^n_{kG}(M,M)$$ show has two distinguished ideals $$I^* \subseteq J^* {\mathcal {E}}_M^*$$ such $$I^*$$ bounded above in degrees, {E}}_M^*/J^*$$ below degree $$J^*/I^*$$ eventually periodic with terms dimension. ...

the proof of the local nilpotence theorem for 4-engel groups was completed by g. havas and m. vaughan-lee in 2005. the complete proof on the other hand is spread over several articles and the aim of this paper is to give a complete coherent linear version. in the process we are also able to make few simplifications and in particular we are able to merge two of the key steps into one.

Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth-Bendix algorithm. In this paper we give a short hand-proof of the nilpotency of T .