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I was trying to understand more about orders of elements in a cyclic group $G$ and non-trivial subgroups that were generated from a cyclic group $G$ and so read the following example: Non trivial subgroups in cyclic group G.

But I do not how one would find trivial and non-trivial groups for say a group $G = \mathbb{Z}^{\times}_{73}$. I get that you have to do $73-1$, as $73$ is prime so you are left with $72$ and then we find all the factors of $72$ which are $\{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}.$

After this, we have to try and find numbers that are not $1 \text{ mod } 72$. I do not where to start in terms of finding the subgroups like do I start with $2^1 \text{ mod } 72, 2^2 \text{ mod } 72, 2^3 \text{ mod } 72, \dots,2^{72} \text{ mod } 72$ to find the subgroups.

Like is there a way of finding non trivial subgroups in a cyclic group?

Jyrki Lahtonen
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user697317
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  • Find elements $g$ with nontrivial order, then $\langle g \rangle$ is a non-trivial subgroup. – Qi Zhu Aug 21 '19 at 08:06
  • @QiZhu how do i do that? what does non trivial even mean, are you able to show an example in the context of Z*73 – user697317 Aug 21 '19 at 08:17
  • Non-trivial means $1 < \operatorname{ord}(g) < |G|$. Finding the order of elements is not easy in general - see e.g. https://math.stackexchange.com/questions/1025578. – Qi Zhu Aug 21 '19 at 08:45
  • @QiZhu I dont suppose you show me one example for my question please? I just want to understand it better – user697317 Aug 21 '19 at 09:39
  • The link I gave you, does exactly this for $41$, it's no different here... – Qi Zhu Aug 21 '19 at 09:45

1 Answers1

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First find a generator $g$ of $G$ i.e. an element of $G$ that has order $|G|$. If $G=\mathbb{Z}_p$ for a prime $p$ then a generator of $G$ is called a primitive root of $p$.

Each element of $G$ can be expressed as $g^k$ for some integer $1 \le k \le |G|$. And $g^k$ will generate a sub-group $<g^k>$ of $G$ with elements $\{g^k, g^{2k}, g^{3k}, \dots \}$. Eventually we reach a multiple of $k$ which is also a multiple of $|G|$ i.e. $nk=m|G|=\text{LCM}(k,|G|)$. At this point we have

$g^{nk}=g^{m|G|}=(g^{|G|})^m=e^m=e$

where $e$ is the identity element in $G$ (we know that $g^{|G|}=e$ because $g$ has order $|G|$). So $<g^k>$ has $n$ elements.

We also know that

$(nk)\text{GCD}(k,|G|)=\text{LCM}(k,|G|)\text{GCD}(k,|G|)=k|G|$

so the order of $<g^k>$ is $\frac{|G|}{\text{GCD}(k,|G|)}$.

For example, if $|G|=72$ then the order of $<g^{15}>$ is $\frac{72}{\text{GCD}(15,72)}=\frac{72}{3}=24$, and the order of $<g^{16}>$ is $\frac{72}{\text{GCD}(16,72)}=\frac{72}{8}=9$.

gandalf61
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  • So in the case of 72, a non-trivial subgroup would be 15? Sorry its just i dont know where the 15 even came from in your example – user697317 Aug 21 '19 at 12:21
  • @user6973173333 First you need to find a primitive root of $73$ - a number with order $72$ in $\mathbb{Z}^{\times}{73}$. Then you raise this primitive root to the power of $15$ modulo $73$. This number will then generate a non-trivial sub-group of $\mathbb{Z}^{\times}{73}$. There is nothing special about $15$ - $g^2$, $g^3$, $g^4$, $g^6$, $g^8$ etc. will all generate non-trivial sub-groups. – gandalf61 Aug 21 '19 at 12:28
  • how would one find a primitive root of 73? So are you saying that <15> is 1 non trivial subgroup and likewise I have to find other subgroups? Also where did the 15 come from? – user697317 Aug 21 '19 at 12:33
  • @user6973173333 I picked the power $15$ at random - it is not important. If you want to understand $\mathbb{Z}^{\times}_{73}$ you should start by reading these Wikipedia articles: Primitive root modulo n, Multiplicative group of integers module n. – gandalf61 Aug 21 '19 at 12:53