# How to prove $\,{\rm order}(a^k) = n/\gcd(n,k)\,$ for $\,n={\rm order}(a)$?

This is an exercise from "Contemporary Abstract Algebra" I'm not sure how to solve.

Exercise: Let $\langle a\rangle$ be a (cyclic) group of order $n$. Prove that the order of $a^k=\frac{n}{\gcd(n,k)}$.

Direction: (1) Let $d=\gcd(n,k)$, thus by the Euclidian algorithm we can find $X,Y\in\mathbb{Z}$ s.t. $d=Xn+Yk$, thus, $a^d=a^{Xn+Yk}=a^{Xn}a^{Yk}=(a^n)^X(a^k)^Y=(a^k)^Y$. What to do from here?

(2) We know that $d|n$, thus $\langle a^{n/d} \rangle$ is of order $d$ and $\langle a^d \rangle$ is of order $\frac{n}{d}=\frac{n}{\gcd(n,k)}$. Is it mean that $d=k$? Where is my mistake?

You have two things to show. Namely that:

1. $(a^k)^{n/\gcd(n,k)}=e$ (where $e$ denotes the identity)

and that $n/\gcd(n,k)$ is the smallest positive power $p$ of $a^k$ such that $(a^k)^p=e$:

1. For all $m>0$: $(a^k)^m=e \implies n/\gcd(n,k)\leq m$

The first part is easy:

$$(a^k)^{n/\gcd(n,k)}=(a^n)^{k/\gcd(n,k)}=e^{k/\gcd(n,k)}=e$$

For the second part, let $m\in\mathbb{N}$ be such that $(a^k)^m=a^{km}=e$. Since the order of $a$ is $n$, it follows that $n\mid km$. Therefore we also have

$$\frac{n}{\gcd(n,k)}\mid \frac{k}{\gcd(n,k)}m$$

Now $\gcd(n/\gcd(n,k),k/\gcd(n,k))=1$ (try to prove this), so it follows that

$$n/\gcd(n,k)\mid m$$

Hence in particular $n/\gcd(n,k)\leq m$.

• What if $n=25$ and $k=30$? then $\gcd(n/\gcd(n,k),k)=5$. Jan 24, 2018 at 14:39
• @Silent Uncountable should have instead wrote $$n\mid km \implies \frac{n}{g}\mid \frac{k}{g}m\implies \frac{n}{g}\mid m,$$ where $g=\gcd(n,k)$, since $\gcd(n/g,k/g)=1$ is true (while $\gcd(n/g,k)=1$ needn't be).
– anon
Jan 25, 2018 at 6:53
• @anon good one! Jan 25, 2018 at 7:58
• – BCLC
Aug 31, 2018 at 17:08

The order of $a^k$ is the smallest $r>0$ such that $a^{kr}=e$, i.e. such that $kr$ is a multiple of the order $n$ of $a$. This means $kr$ is the least common multiple of $k$ and $n$.

As $\operatorname{lcm}(k,n)=\dfrac{kn}{\gcd(k,n)}$, we have: $\quad r=\dfrac{n}{\gcd(k,n)}.$

• – BCLC
Aug 31, 2018 at 17:09
• Do you mean corollary 2.8.10? $x^k$ is a element of the group generated by $x$, which has order $n$. But this result is more precise than the corollary. Aug 31, 2018 at 17:39
• Bernard, interesting, but no. I really mean Cor 2.8.11 intending to use $\gcd(k,p)=1$ for any or all $0 < k < p$. Like somehow we might show that the orders of the non-identity elements are given by $p/\gcd(k,p)$, which of course simplifies to $p$.
– BCLC
Aug 31, 2018 at 17:46
• It might be used. However, I wonder whether the notion of order of an element doesn't ultimately on Lagrange's theorem. Aug 31, 2018 at 19:12
• Thanks Bernard!
– BCLC
Aug 31, 2018 at 19:17

You can think additively in $$\Bbb Z/n\Bbb Z$$. Say $$\bar a$$ a generator of $$\Bbb Z/n\Bbb Z$$, whence $$\gcd(a,n)=1$$. The order of $$k\bar a \in \Bbb Z/n\Bbb Z$$ is by definition the least positive integer, say $$o(k\bar a)$$, such that $$o(k\bar a)k\bar a=\bar 0$$. But, $$o(k\bar a)k\bar a=\overline{o(k\bar a)ka}$$, and hence:

\begin{alignat}{1} o(k\bar a)k\bar a=\bar 0 &\iff o(k\bar a)ka \equiv 0 \pmod n \\ &\iff o(k\bar a)ka=mn \end{alignat}

for some $$m\in \Bbb Z$$. Therefore, the order of $$k\bar a$$ is the least positive integer of the form ($$*$$) $$\frac{n}{(ka/m)}$$, and it is then gotten when the denominator in ($$*$$) is the greatest divisor of $$ka$$ which is also divisor of $$n$$, namely:

\begin{alignat}{1} o(k\bar a) &= \frac{n}{\gcd(ka,n)} \\ &= \frac{n}{\gcd(k,n)} \\ \end{alignat} where the last equality follows from $$\gcd(a,n)=1$$.

The proof is simpler (and more $$\rm\color{darkorange}{general}$$) using basic gcd / lcm laws (vs. Bezout gcd identity).

Lemma $$\ \ (a^{\large k})^{\large j}\! = 1\color{#90f}{\underset{a^n\,=\,1}\Longleftarrow}\!\!\!\!\!\color{#0a0}{\overset{{\rm ord}\,a\,=\,n}{\Longrightarrow}} n/(n,k)\mid j\,.\ \$$ Proof:  immediate by Euclids Lemma, i.e.

$$(a^{\large k})^{\large j}\! = 1\color{#90f}{\underset{a^n\,=\,1}\Longleftarrow}\!\!\!\!\!\color{#0a0}{\overset{{\rm ord}\,a\,=\,n}{\Longrightarrow}} n\mid kj\!\iff\! n\mid nj,kj\!\color{#c00}\iff\! n\mid(nj,kj)\!=\!(n,k)j\!\iff\! n/(n,k)\mid j\qquad$$

$$\,\color{#0a0}{\rm First\ (\Rightarrow)}\,$$ is by $$\,n = {\rm\ ord}\, a,\,$$ $$\rm\color{#c00}{third}$$ by the definition/universal property of the gcd and the gcd distributive law. $$\,\color{#90f}{\rm Note}$$ that the proof of direction $$\color{#90f}{(\Leftarrow)}$$ needs only $$\,\color{#90f}{a^n =1},\,$$ not $$\,\color{#0a0}{{\rm ord}\,a = n}$$.

Alternatively, dually, we can use lcm instead of gcd, using notation $$\,[x,y] :={\rm lcm}(x,y),$$

$$(a^{\large k})^{\large j}\! = 1\color{#90f}{\underset{a^n\,=\,1}\Longleftarrow}\!\!\!\!\!\color{#0a0}{\overset{{\rm ord}\,a\,=\,n}{\Longrightarrow}} n\mid kj\iff n,k\mid kj\iff [n,k]\mid kj\iff [n,k]/k\mid j\qquad\qquad$$

Both proofs are equivalent by $$\ [n,k]/k = n/(n,k),\$$ i.e. $$\ [n,k](n,k) = nk\$$ [gcd $$*$$ lcm law]

Corollary $$\ \ \bbox[5px,border:1px solid #c00]{{\rm ord}(a) = n\,\Rightarrow\,{\rm ord}(a^{\large k}) = \dfrac{n}{(n,k)} = \dfrac{[n,k]}k}\$$ since, generally

$$\qquad\qquad\quad\underbrace{ b^{\large j} = 1\iff i\mid j}_{\small \textstyle \text{Lemma has }\, b = a^k}\$$ is equivalent to $$\,b\,$$ has order $$\,i.\,$$

$$\rm\color{darkorange}{Generality\!\!:}$$ proofs using gcd laws (vs. Bezout) apply much more generally, e.g. in UFDs where Bezout fails, e.g. $$\,\Bbb Z[x]\,$$ and $$\,\Bbb Q[x,y],\,$$ e.g. $$\,(x,2) = 1 = (x,y)\,$$ but the gcds cannot be written as linear combinations, else $$\, 1 = x\,f(x,y) + y\,g(x,y)\Rightarrow\,1=0\,$$ by evaluating at $$\,x=0=y.\,$$