What is the connection between L and gcd(k,n) in cyclic subgroups?

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Discussion Overview

The discussion revolves around the relationship between the order of an element in a cyclic group and the greatest common divisor (gcd) of two integers, specifically in the context of cyclic subgroups. Participants explore the implications of the formula L = n/gcd(k, n) and seek clarification on its derivation and significance.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the connection between L and gcd(k, n), particularly how the formula L = n/gcd(k, n) is derived.
  • Another participant proposes that L is the smallest integer such that kL is a multiple of n, suggesting that if gcd(k, n) = 1, then L equals n.
  • Concerns are raised about the clarity of the original proof, particularly regarding the role of the integer b in the equation a^kL = a^bn.
  • Some participants question the generalization of the equation a^kL = a^bn = 1 to find L, indicating a lack of consensus on the implications of the proof.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the proof and its implications. There is no consensus on the clarity of the proof or the interpretation of the variables involved, particularly b.

Contextual Notes

Participants note the potential ambiguity in the definitions and roles of variables in the proof, as well as the reliance on the properties of gcd without fully resolving the implications for L.

SticksandStones
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Today we learned about subgroups of cyclic groups G = <a>. During the discussion we reached this point:

|<a^k>| = minimum L, L > 0, such that a^(kL) = 1.
|G| = n.
Then a^kL = a^bn, thus kL = bn, and thus L = n/gcd(k, n).

However, I don't understand the bolded. My number theory is terrible, and I don't really see where the gcd(k, n) comes from.

I understand that if gcd(k, n) = 1 that <a^k> = <a>, but the connection to L = n/gcd(k,n) just isn't apparent to me. Can someone shine some light on this for me?


Thanks, I appreciate it!
 
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Bleh, I think I figured it out. kL is a multiple of n and L is the lowest integer that makes kL a multiple of N. If gcd(k,n)=1 then L is just n, otherwise it has to be a multiple of n. If gcd(k,n)=d then k=sd and L=n/gcd(k,n) so kL=sn and s=b.

Is this correct?
 
I really don't like the way the original proof is worded. What exactly is b?
 
morphism said:
I really don't like the way the original proof is worded. What exactly is b?

Some arbitrary integer as far as I can tell.
 
If that's the case, then a^kL = a^bn doesn't in general imply that kL = bn.
 
Ok then I'm really confused. a^kL=a^bn = 1, how can I then generalize this to find L?

EDIT: Normally I'd just ask the professor, but it was a "substitute professor" if you will and I honestly don't know who the man is/was.
 

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