Conditions are given for a class 2 nilpotent group to have no central extensions of class 3. This is related to Betti numbers and to the problem of representing a class 2 nilpotent group as the fundamental group of a smooth projective variety. Surveys of the work on the characterization of the fundamental groups of smooth projective varieties and Kähler manifolds (see [1],[3], [9]) indicate tha...

We exhibit the primitive central idempotents of a semisimple group algebra of a finite nilpotent group over an arbitrary field (without using group characters), examining the abelian case separately. Our result extends and improves the main result in [1].

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full G-shift for a finitely-generated torsion-free nilpotent group G. Using bounds for the Möbius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape

‎Let $G$ be a finite group and $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$‎. ‎In this paper‎, ‎all nilpotent groups $G$ with $nu(G)=3$ are classified‎.  

For a finite group G, let W(G) denotes the set of orders elements G. In this paper we study jW(G)j and show that cyclic order n has maximum value among all groups same order. Furthermore notion in nilpotent non-nilpotent state some inequality for it. Among result minimum is power 2 or it pertains to group.

Let G be a complex symplectic group. We construct a certain Gvariety N, which shares many nice properties with the nilpotent cone of G. For example, it is normal, has finitely many G-orbits, and admits a G-equivariant resolution of singularity in a similar way to the Springer resolution. Our new G-variety admits an action of G× C × C instead of the G×C-action on the nilpotent cone. This enables...

Let G be a group. An endomorphism φ of G is called rational if there exist a1, . . . , ar ∈ G and h1, . . . , hr ∈ Z, such that φ(x) = (xa1)1 . . . (xar)r for all x ∈ G. We denote by Endr(G) the group of invertible rational endomorphisms of G. In this note, we prove that G is nilpotent of class c (c ≥ 3) if and only if Endr(G) is nilpotent of class c − 1. Mathematics Subject Classification: 20E...

The spectrum of a Gelfand pair (K nN, K), where N is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz K-invariant functions on N . We also show the converse in the case of the Gelfand pair (SO(3) n N3,2, SO(3)), where N3,2 is the free two-step nilpotent Lie group with three generators...

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, the nilpotent case. It turns out that context, like for finite nilpotency is equivalent to normalizer property or uniqueness of Sylow subgroups, provided maximal normal torsion-free subgroup nilpotent. As consequence, we show decompositions prove Lie group an expansion reals if...

Let P be an arbitrary set of primes. The P -nilpotent completion of a group G is defined by the group homomorphism η : G → G P̂ where G P̂ = invlim(G/ΓiG)P . Here Γ2G is the commutator subgroup [G,G] and ΓiG the subgroup [G,Γi−1G] when i > 2. In this paper, we prove that P -nilpotent completion of an infinitely generated free group F does not induce an isomorphism on the first homology group with...