Group Generators: Showing a^r is Generator of G

In summary, the homework statement is that for any generator a of G, there exists an integer m such that a^mr=b. This integer must also be relatively prime to n.
  • #1
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Homework Statement



Let G be a finite cyclic group of order n. Let a be a generator. Let r be an integer, not zero, and relatively prime to n.

a)Show that a^r is also a generator of G.


Homework Equations





The Attempt at a Solution



For the first one I need to show that for any element b in G there exists some integer m s.t. (a^r)^m = b. I have no idea what to do. The great thing about this is that I will be tested on this tomorrow. Ha, you get it. Pleast try to answer as soon as possible. Thanks for any help.
 
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  • #2
b=a^s, since a generates G. Can you solve r*m=s mod n, where r is relatively prime to n. Hint: does r have an inverse in Z(n)?
 
  • #3
Thanks. But we haven't gotten around Z(n) yet. Don't know what that is. Not sure I'm following you either. Does the solution require knowledge of Z(n) or is there any other way around it? Thanks again.
 
  • #4
You need to know that if r and n are relatively prime, then there is an integer t such that t*r=1 mod n.
 
  • #5
oh, so now we multiply both sides of the eqn by s:

s*t*r = s mod n

and let m = s*t. Now we get a^(str) = a^(mr) = a^(s mod n) = a^s.

We have shown for any b in G s.t. b = a^s for some s, b = a^rm for some m.

Is that good?
 
  • #6
Wouldn't we have to show also that the period of a^r is n, to make sure that a^r is not generating something with greater size than G?
 
  • #7
Oh, because a^r is an element of G, it can only generate subgroups which cannot have greater order than n. ok its clear. Thanks a lot.
 

1. What is a group generator?

A group generator is an element in a group that can generate all the other elements in the group through repeated multiplication. In other words, it is an element that when raised to different powers, can produce all the elements in the group without any repetitions.

2. What is the significance of showing a^r is a generator of G?

If we can show that a specific element, a^r, can be a generator of a group G, it means that we can generate all the elements in G using only a single element. This simplifies calculations and proofs in group theory and can help identify the structure of a group.

3. How do you prove that a^r is a generator of G?

To prove that a^r is a generator of a group G, we need to show that raising a^r to different powers can produce all the elements in G. This can be done through a proof by contradiction, where we assume that there is an element in G that cannot be generated by a^r, and then show that this leads to a contradiction.

4. Can any element in a group be a generator?

No, not all elements in a group can be generators. In order for an element to be a generator, it must be relatively prime to the order of the group. This means that the only common divisor between the element and the order of the group is 1.

5. How is the order of a group related to its generators?

The order of a group is the number of elements in the group. The number of generators in a group is related to its order through Euler's totient function, where the number of generators is equal to phi(n), where n is the order of the group. In other words, the number of generators is equal to the number of elements in the group that are relatively prime to the group's order.

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