TITLE:
A Represent of Generators of the Cyclic Group of Higher Even, Odd and Prime Order for Composition Being Multiplication
AUTHORS:
Faruque Hossain Mozumder, Md. Sahidul Islam, Md. Mehedi Hasan Modern, Md. Abdul Mannan
KEYWORDS:
Order of Element of a Group, Multiplication Composition, Highest Common Factor, Generator
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.12,
December
26,
2024
ABSTRACT: This paper aims to treat a study of generators of the cyclic group of higher even, odd, and prime order for composition being multiplication. In fact we developed order of a group, order of element of a group and generators of the cyclic group in real numbers. Also we express cyclic and generators of the group for composition in real numbers. Here we discuss the higher order of groups in different types of order, and generators of the cyclic group which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element
a
m
∈G
in which
a
n
=e
= identity element, then find Highest Common Factor i.e. (H.C.F) of
m
and
n
. When
G
is a finite group, every element must have finite order but the converse is false. There are infinite groups where each element has a finite order. There may be more than one generator of a cyclic group. Also every cyclic group is necessarily abelian. But show that every infinite cyclic group contains only two generators. Finally, we find out the generators of the cyclic group of higher even, odd and prime order in different types of the group for composition being multiplication.