We give a classification of two-generator p-groups of nilpotency class 2. Using this classification, we give a formula for the number of such groups of order p n in terms of the partitions of n of length 3, and find formulas for the number and size of their conjugacy classes.

Abstract. The order of an element x in a finite group G is the smallest positive integer k, such that k x is the group identity. The set of all possible such orders joint with the number of elements that each order referred to, is called the order classes of G. The conjugacy classes of dihedral groups already known, the conjugacy classes is a refinement partition to the order classes. In this paper, the order classes of dihedral groups are derived. In addition, clarifications for some cases related to the size of the groups were given.

As a system of abstract algebra, evolution algebras are commutative and non-associative algebras. There is no deep structure theorem for general non-associative algebras. However, there are deep structure theorem and classification theorem for evolution algebras because it has been introduced concepts of dynamical systems to evolution algebras. Recently, in [25], it has been studied some properties of nilpotent evolution algebra with maximal index (dim E2 = dim E − 1). This paper is devoted to studying nilpotent finite-dimensional evolution algebras E with dim E2 =dim E − 2. We describe Lie algebras related to the evolution of algebras. Moreover, this result allowed us to characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras are found.

No abstract

In this paper, we consider [Formula: see text]-dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra.