A characterization of groups of exponent   which are nilpotent of class at most 2

doi:10.1285/i15900932v30n2p149

A characterization of groups of exponent $p$ which are nilpotent of class at most 2

Domenico Lenzi

Abstract

Let $(\mathbf{G},+)$ be a group of prime exponent $p = 2n + 1$ . In this paper we prove that $(\mathbf{G},+)$ is nilpotent of class at most 2 if and only if one of the following properties is true:
$i)$ $\mathbf{G}$ is also the support of a commutative group $(\mathbf{G},+')$ such that $(\mathbf{G},+)$ and $(\mathbf{G},+')$ have the same cyclic cosets [cosets of order $p$ ]. $ii)$ the operation $\oplus$ defined on $\mathbf{G}$ by putting $x \oplus y = x/2 + y + x/2$ , gives $\mathbf{G}$ a structure of commutative group.\end

DOI Code: 10.1285/i15900932v30n2p149

Keywords: nilpotent groups; group partition

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This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.

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Note di Matematica

A characterization of groups of exponent which are nilpotent of class at most 2

Abstract

A characterization of groups of exponent $p$ which are nilpotent of class at most 2