In this paper some results on the structure of finite-dimensional Lie algebras are obtained by means of the concept of maximal abelian dimension. More concretely, a sufficient condition is given for the solvability in finite-dimensional Lie algebras by using maximal abelian dimensions. Besides, a necessary condition for the nilpotency is also stated for such Lie algebras. Finally, the maximal a...

The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f : H → K which fixes the identity and such that h commutes with its image h for all h ∈ H. The present paper gives support to conjectures about the ni...

Let G be a nite group and let i (G) denote the proportion of ordered pairs of G that generate a subgroup of nilpotency class i. Various properties of the i 's are established. In particular it is shown that i = k i jGj=jGj 2 for some non-negative integers k i and that P 1 i=1 i is either 1 or at most 1/2 for solvable groups.

By studying the group of self homotopy equivalences of the localization (at a prime p and/or zero) of some aspherical complexes, we show that, contrary to the case when the considered space is a nilpotent complex, E # (Xp) is in general different from E # (X)p . That is the case even when X = K(G, 1) is a finite complex and/or G satisfies extra finiteness or nilpotency conditions, for instance,...

We have obtained the nilpotent BRST symmetry of Yang-Mills theory restricted to the Gribov region. In fact, we give a modified BRST transformation that leaves the Gribov-Zwanziger action invariant and obeys the nilpotency, although the usual BRST symmetry is broken due to the non-vanishing Gribov parameter representing the presence of the Gribov horizon in the configuration space of the Yang-Mi...

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We introduce the notion of N-topo nilpotent fuzzy set in a fuzzy neighborhood ring and develop some fundamental results. Here we show that a fuzzy neighborhood ring is locally inversely bounded if and only if for all 0 < α < 1, the α-level topological rings are locally inversely bounded. This leads us to prove a characterization theorem which says that if a fuzzy neighborhood ring on a division...