Moderate# Order of Elements in Abelian Groups

ABSALG-EWKO1Z

Let $C_m$ be a finite cyclic group of order $m\ge 1$. Therefore $|C_m|=m$ and there is an element $g\in C_m$ such that, in multiplicative notation, we have:

$$C_m=\{g^n\mid 0\le n\le m-1\}$$

We say that $g$ generates $C_m$ and we write $C_m=\langle g\rangle$.

Let $G$ be a group with identity element $e$. An element $h\in G$ has order $k\ge 1$, denoted $|h|=k$, if $|\langle h\rangle|=k$. Therefore $k$ is the smallest positive integer with $h^k=e$.

An abelian group $G$, where we write the composition law using the multiplicative notation, is one such that $ab=ba$ for every $a,b\in G$. Notice that a finite cyclic group $C_m$ is abelian.

Which of the following is **ALWAYS** true?