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

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In summary, we learned about subgroups of cyclic groups G = <a> and discussed the minimum L needed for |<a^k>| to equal L, where L is greater than 0 and a^(kL) = 1. We also learned that |G| = n and a^kL = a^bn, leading to the conclusion that L = n/gcd(k, n). This is because if gcd(k, n) = 1, then <a^k> = <a>, but if gcd(k, n) = d, then L must be a multiple of n. However, the connection between kL and bn is not clear and may require further clarification. Additionally, the variable b is
  • #1
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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  • #2
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?
 
  • #3
I really don't like the way the original proof is worded. What exactly is b?
 
  • #4
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.
 
  • #5
If that's the case, then a^kL = a^bn doesn't in general imply that kL = bn.
 
  • #6
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.
 

1. What is the definition of a cyclic subgroup?

A cyclic subgroup is a subset of a group that is generated by a single element, known as a generator. This means that every element in the subgroup can be expressed as a power of the generator, and the subgroup will contain all possible powers of the generator.

2. How can I identify the order of a cyclic subgroup?

The order of a cyclic subgroup can be determined by finding the smallest positive integer k such that the generator raised to the power of k results in the identity element of the group. This value of k is known as the order of the generator and is equal to the order of the cyclic subgroup.

3. Can a cyclic subgroup have a different order than its generator?

Yes, a cyclic subgroup can have a different order than its generator. This is because the order of a subgroup is the number of elements it contains, while the order of a generator is the smallest positive integer that results in the identity element. Therefore, it is possible for the order of a generator to be smaller than the order of the subgroup it generates.

4. How does the order of cyclic subgroups relate to the order of the larger group?

The order of a cyclic subgroup must be a divisor of the order of the larger group. This is because the subgroup contains a subset of the elements of the larger group, and the order of the subgroup must divide the order of the larger group in order for the subgroup to be a proper subgroup.

5. Can two different cyclic subgroups have the same order?

Yes, it is possible for two different cyclic subgroups to have the same order. This can happen when two different generators in the larger group have the same order, resulting in two different cyclic subgroups with the same number of elements. However, the elements within these subgroups will not necessarily be the same.

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