In "Mathematical Methods for Physics and Engineering" by Riley, Hobson, and Bence, in the chapter on group theory, the order of the element $X$ in the finite group $G$ is the number $m$ such that $X^m=I$; thus, the order of the identity is always $1$. A cyclic group is then defined as a group generated from a single element $X$, i.e.,
$$G=\{I,X,X^2,X^3,\dots,X^{g-1}\},$$
where $g$ is the order of the group. They then write that
It is clear that cyclic groups are always Abelian and that each element, apart
from the identity, has order g, the order of the group itself.
I don't see why the second statement is true; for example, when $g=4$, then the element $X^2$ should have order $2$. Am I wrong?
Update: I reached out to the first author and he said that this error was corrected in the $18$th reprint of the $3$rd edition (I was looking at the $2$nd edition).
Best Answer
You are right.
The main problem is in the given definition of order. In fact, the order $|a|\in\Bbb N$ of an element $a$ in a group is the smallest positive integer such that $a^{|a|}=e$.
Consider the cyclic group $\Bbb Z_6$ of six elements. Let $X$ be a generator. Then $X^2$ has order three and $X^3$ has order two.