Question

Show that any cyclic group is Aeolian. Hint: Does a matrix commute with itself?

Short answer

It is verified that any cyclic group is Aeolian

Step-by-step solution

Given information

The given information is A cyclic group.

Definition of Cyclic Group

A cyclic or monogenous group is a group produced by a single element in group theory, a subfield of abstract algebra. It consists of an element gsuch that all other members of the group may be derived by repeatedly applying the group operation to gor its inverse. In other words, it is a set of invertible elements with a single associative binary action. In either multiplicative or additive notation, each element can be expressed as a power of gor as a multiple of g. The term "generator of the group" refers to this element, g.

Verify the given statement

In this problem, we show that any cyclic group is Abelian. A cyclic group of order n is given by the elements

$A,A^2,...A^{n-1},A^n=1$

Since each element can be written as a power of a single element, we know that multiplication of any two elements commutes (it is of the form $A^i{A}^j=A^j{A}^i$).

Hence, We have shown that any cyclic group is Abelian.