How can $1=\frac{\operatorname{ord} a}{\gcd(\operatorname{ord}a,k)}\mid\frac{k}{\gcd(\operatorname{ord}a,k)}$?

Question

Let $G$ be a group and $a\in G$ an element with finite order. For any $k\in\mathbb{Z}$,

$\operatorname{ord}a^k=\frac{\operatorname{ord} a}{\gcd(\operatorname{ord}a,k)}$

ord-order ,gcd-greatest common divisor

Proof:

Consider $k,m\in\mathbb{Z}$ such that:

$$(a^k)^m=e\implies a^{km}=e\implies \operatorname{ord} a \mid km\implies \frac{\operatorname{ord}a}{\gcd(\operatorname{ord}a,k)}\mid\frac{k}{\gcd(\operatorname{ord}a,k)}m$$

Therefore:

$$1=\frac{\operatorname{ord}a}{\gcd(\operatorname{ord}a,k)}\mid\frac{k}{\gcd(\operatorname{ord}a,k)}\implies\frac{\operatorname{ord} a}{\gcd(\operatorname{ord}a,k)}|m$$

So we have:

$$\operatorname{ord}a^k=\frac{\operatorname{ord} a}{\gcd(\operatorname{ord}a,k)}$$

Question:

I am struggling with this material I am reading, in the sense I do not understand why $1=\frac{\operatorname{ord} a}{\gcd(\operatorname{ord}a,k)} \mid \frac{k}{\gcd(\operatorname{ord}a,k)}$ is true. I have been looking in books but I found nothing of this. How can $1=\frac{\operatorname{ord} a}{\gcd(\operatorname{ord}a,k)}\mid\frac{k}{\gcd(\operatorname{ord}a,k)}$? The least common divisor between two numbers is not necessarily equal to the first or second. $\frac{15}{3}$ and $\frac{20}{3} \neq 1$ and $3=\gcd(15,20)\neq 1$.

If you find problems with terminology used, please leave a comment.

Thanks in advance!

Answer 1

You may find the following easier to understand.

Let $|a|$ represents the order of $a$ and let the g.c.d. of $|a|$ and $k$ be $g$.

$(a^k)^\frac{|a|}{g}=(a^{|a|})^\frac{k}{g}=e$ and so $|a^k|$ divides $\frac{|a|}{g}$.

Conversely, $(a^k)^{|a^k|}=e$ implies $a^{k|a^k|}=e$ and so $|a|$ divides $k|a^k|$. Then $\frac{|a|}{g}$ divides $\frac{k|a^k|}{g}$ and, since $\frac{|a|}{g}$ and $\frac{k}{g}$ are coprime, $\frac{|a|}{g}$ must divide $|a^k|$.

Therefore $\frac{|a|}{g}=|a^k|$, as you required.