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Problem statement:

Let $p_1$ and $p_2$ be two distinct prime numbers where $p_2 < p_1$. I want to prove that if $g \in \mathbb{Z}_{p_1}^{*}$ has the property that $g^{p_2} \equiv 1 \mod p_1$, then the set $S = \{g^{i} \bmod p_1 \mid 0 \leq i < p_2\}$ is closed under multiplication (i.e., $S$ is a subgroup of $\mathbb{Z}_{p_1}^{*}$).

Solution attempt:

By Lagrange's theorem we know that the order of any proper subgroup $H$ must divide the order of the finite group $G$. We also know that the order of $g$ is defined as the smallest positive integer $d$ such that $$ g^d \equiv 1 \mod p. $$ So, my idea is that we need to show that the order of $S$ divides the order of $\mathbb{Z}_{p_1}^* = \phi(p_1) = p_1 - 1$.

For the elements $g \in \mathbb{Z}_{p_1}^*$ that have the property that $g^{p_2} \equiv 1 \mod p_1$ we know by the definition of the order of $g$ that the order for the cyclic subgroup $S$ generated by $g$ must be $p_2$.

Here is where I get confused...

Question(s):

  1. We know that $|S| = p_2$. For $S$ to be a proper subgroup of $\mathbb{Z}_{p_1}^*$, $p_2$ must divide the order of the group which is $p_1 - 1$. It is clear that this is not always the case. I've gone wrong and I don't know how to proceed.
Arturo Magidin
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KSI
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  • You are going down the wrong path. $p_1$ plays no role here (other than the fact that $\Bbb Z_{p_1}^*$ is a monoid, i.e. closed under multiplication). Similarly primality is not relevant. For $,g,$ in any monoid, if $,g^k = 1,$ for $,k\ge 1,$ then $,S = {1,g,g^2,\ldots,g^{k-1}},$ is closed under multiplication simply because a product has form $,g^n,,$ and $,\color{#c00}{g^k=1}\Rightarrow g^n = g^{n\bmod k},,$ since by division $,n = kq+r,\ 0\le r\le k-1,,$ so $, g^n = (\color{#c00}{g^k})^q g^r = \color{#c00}{1}^kg^r = g^r.\ \ $ – Bill Dubuque Nov 28 '24 at 19:40
  • Above is a simply a special case of ubiquitous mod order reduction. Note also $,(g^i)^{-1} = g^{k-i},$ by $,g^{k-i}g^i = g^k = 1.\ \ $ – Bill Dubuque Nov 28 '24 at 19:41
  • Re: your wrong path: generally in any group of order $,n,,$ if $,g^k = 1,$ and $,k,$ is coprime to $,n,$ then $,g = 1,,$ since - by Lagrange - the order of $,g,$ divides the coprimes $,k,n,$ so it must be $1$, hence $,S = \langle g\rangle = {1},$ (which is trivially closed under multiplication). This is what happens in your case when the prime $,p_2\nmid p_1-1.,$ So nothing is "wrong" in this case. But none of this is relevant to your original problem. – Bill Dubuque Nov 28 '24 at 19:56

1 Answers1

2

You can get it simpler.

Since $p_2<p_1$ :

You get with any $(i,j) \in [|0,p_2|[$ :

$$ g^{i+j} \mod p_1 $$

Two cases :

  • $i+j\geq p_2$, you know $g^{p_2}=1 \mod p_1$, so you get $g^{i+j}=g^{i+j \mod p_2} \mod p_1 $ and so you get back in $S$.
  • $i+j<p_2$ , so you're in $S$.

In order to be a subgroup, you have to proove that :

$$ (a,b) \in S \implies a \times b^{-1}\in S $$

In other words, taken $(i,j)$ above gives :

$$ g^{i}g^{-j} \mod p_1 $$

So that we want $(i-j \mod p_2) \in [|0,p_2|[ $, which is the case always.

So the group $S$ is a subgroup relatively to $Z_{p_1}^*$.

But $|S|\neq p_2$ in general. Because $p_2$ may not be the minimal order of the element $g$.

EDX
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  • More clearly viewed as a special case of mod order reduction - see my comments on the question. – Bill Dubuque Nov 28 '24 at 20:01
  • In the first part you prove the property that is "Closure". In the second part you seem to be doing something with an inverse, but I don't exactly understand why? Are you trying to prove the existence of an inverse element and that multiplying by that yields the identity element $e=1$ in this case? Associativity holds as the group operation is multiplication, identity element exists as per definition of $S$, closure you have proven, so what is left to prove is the existence of an inverse element for all $g \in \mathbb{Z}_{p_1}^*$, right? – KSI Dec 07 '24 at 18:32
  • I prove first closure, as you want to do in your OP. Then I precise that in order to prove that a group is a subgroup you just have to prove it contains neutral, isn't empty so, and $a.b^{-1} \in S$ when $(a,b) \in S^2$. So I prove this is closed under multiplication, is indeed a subgroup and then because Lagrange theorem hold, it proves that the minimal order of $g$ is less than $p_2$. Hope this clarify. – EDX Dec 08 '24 at 13:20