Maximal Order of an NG-group

TL;DR

This paper investigates NG-groups, which contain no bijections, and establishes that their maximal order is at most n, by showing they are isomorphic to permutation groups on quotient sets.

Contribution

It provides a new characterization of NG-groups and determines their maximal order, linking them to permutation groups on quotient sets.

Findings

Maximal order of NG-groups is at most n

NG-groups are isomorphic to permutation groups on quotient sets

Established a bound on the size of NG-groups

Abstract

This study was aimed to consider the NG-group that consisting of transformations on a nonempty set A has no bijection as its element. In addition, it tried to find the maximal order of these groups. It found the order of NG-group not greater than n. Our results proved by showing that any kind of NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A.