# Let $a=b^k$. Prove that $\langle a\rangle=\langle b\rangle$ iff $\gcd(k, \operatorname{ord}(a))=1$.

This is from Pinter's Book of Abstract Algebra Chapter 11, Exercise D5.

Let $$n=\operatorname{ord}(a)$$.

I think I can prove the $$\Rightarrow$$ direction: since $$a^r$$ generates $$\langle a \rangle$$ iff $$\gcd(r, n)=1$$, hence $$a=b^k$$ generates $$\langle b \rangle$$. Since $$a$$ generates $$\langle a\rangle$$ and $$\langle b \rangle$$, the equality holds.

But I have trouble proving the $$\Leftarrow$$ direction. What I can see so far:

$$\langle a\rangle\subseteq\langle b\rangle$$ because $$a = b^k\tag 1$$

$$n\mid\operatorname{ord}(b)\tag 2$$ because the order of cyclic subgroup $$A$$ of cyclic group $$B$$ divides the order of cyclic group B. $$\operatorname{ord}(b)\mid k n\tag 3$$ because $$b^{kn}=a^k=e$$

Because of $$(1)$$, we know that $$\langle a \rangle=\langle b\rangle$$ iff $$a$$ and $$b$$ have the same order, so it feels like I just need to tighten $$(2)$$ and $$(3)$$, but I am stuck. Any help will be appreciated.

• Related $(1)$ Jun 5, 2020 at 6:20
• Related $(2)$. Jun 5, 2020 at 6:29
• Wow, thanks for the edits! Jun 5, 2020 at 7:47
• I put the links in case some other idea occurs to you. ðŸ˜€Those might be helpful in future. Jun 5, 2020 at 8:01
• Also, if you want "$\mid$" for divides , use \mid and, if you want $\nmid$, use \nmid. ðŸ˜€ Jun 5, 2020 at 8:12

Counter example: Consider the cyclic subgroup of $$\mathbb{Z}_{10}$$. Let's say $$a = 2$$ and $$b = 1$$. We know that $$a = b^2$$, $$\operatorname{ord}(a) = 5$$, and $$\gcd(2, 5) = 1$$. However, $$\langle 1 \rangle = \mathbb{Z}_{10}$$ but $$\langle 2 \rangle$$ consists of all even numbers of $$Z_{10}$$. Hence $$\langle a \rangle \neq \langle b \rangle$$.
• I recently asked this same question. Can I assume that the question from Pinter's book has a typo since the converse doesn't hold? I.e. it is not the case that $\langle a\rangle = \langle b\rangle$ whenever $a = b^k$ and the order of $a$ and $k$ are relatively prime? Oct 23, 2023 at 22:41