We investigate the Poisson algebras, in which n-th hypercenter (center) has a finite codimension. It was established that, this case, algebra P includes finite-dimensional ideal K such that P/K is nilpotent (Abelian). Moreover, if of over some field codimension, and does not contain zero divisors, then Abelian.

We associate a graph NG with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of NG and its induced subgraph on G\nil(G), where nil(G) = {x ∈ G | 〈x, y〉 is nilpotent for all y ∈ G}. For any finite group G, we prove that NG has eithe...

The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to g in G. The significance of elements of prime power index is best recognized once one remembers Wielandt's Theorem that elements whose order and index are powers of the same prime p are contained in a normal subgroup of order a power of p and Burnside's Theorem asserting the absence of elements of pr...

The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to g in G. The significance of elements of prime power index is best recognized once one remembers Wielandt's Theorem that elements whose order and index are powers of the same prime p are contained in a normal subgroup of order a power of p and Burnside's Theorem asserting the absence of elements of pr...