The power graph P(G) of a group G is with vertex set G, where two vertices u and v are adjacent if only u≠v um=v or vm=u for some positive integer m. In this paper, we raise study the following question: For which natural numbers n every groups order isomorphic graphs isomorphic? particular, it proved that all such odd number cube-free also they not multiples 16 in general. Moreover, show finit...

The punctured power graph P (G) of a finite group G is the graph which has as vertex set the nonidentity elements of G, where two distinct elements are adjacent if one is a power of the other. We show that P (G) has diameter at most 2 if and only if G is nilpotent and every Sylow subgroup of G is either a cyclic group or a generalized quaternion 2-group. Also, we show that if G is a finite grou...

While powers of the adjacency matrix of a finite graph reveal information about walks on the graph, they fail to distinguish closed walks from cycles. Using elements of an appropriate commutative, nilpotentgenerated algebra, a “new” adjacency matrix can be associated with a random graph on n vertices and |E| edges of nonzero probability. Letting Xk denote the number of k-cycles occurring in a r...

where g±d 6= 0. The positive integer d is called the depth of this Z-grading, and of the nilpotent element e. This notion was previously studied e.g. in [P1]. An element of g of the form e+ F , where F is a non-zero element of g−d, is called a cyclic element, associated with e. In [K1] Kostant proved that any cyclic element, associated with a principal (= regular) nilpotent element e, is regula...

In this paper, we define hedge operation on a residuated skew lattice and investigate some its properties. We get relationships between some special sets as dense, nilpotent, idempotent, regular elements sets and their hedges.  By examples, we show that hedge of a dense element is not a dense and hedge of a regular element is not a regular. Also hedge of a nilpotent element is a nilpotent and h...

Mal’cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically pre...

This paper describes an algorithm for computing maximal tori of the reductive centralizer of a nilpotent element of an exceptional complex symmetric space. It illustrates also a good example of the use of Computer Algebra Systems to help answer important questions in the field of pure mathematics. Such tori play a fundamental rôle in several problems such as: classification of nilpotent orbits ...

A semigroup is nilpotent of degree 3 if it has a zero, every product of 3 elements equals the zero, and some product of 2 elements is non-zero. It is part of the folklore of semigroup theory that almost all finite semigroups are nilpotent of degree 3. We give formulae for the number of nilpotent semigroups of degree 3 on a set with n ∈ N elements up to equality, isomorphism, and isomorphism or ...

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincrea...