# Abstract Algebra - Cyclic groups

(This is my first post on PF btw - I posted on this another thread, but I'm not sure if I was supposed to)

I was doing some practice problems for my exam next week and I could not figure this out.

1. The problem statement, all variables and given/known data

Suppose a is a group element such that |a^28| = 10 and |a^22| = 20. Determine |a|.

2. Relevant equations

Let a be element of order n in group and let k be a positive integer. Then <a^k> = <a^gcd(n,k)> and |a^k| = n/gcd(n,k).

3. The attempt at a solution

10 = n/gcd(n, 28); 20 = n/gcd(n, 22)

Setting n equal to each other, 10gcd(n, 28) = 20gcd(n,22)

gcd(n, 28) = 2gcd(n, 22)

The possible values for n are 4, 8, 12, 16, 20, 24, ... , so on.

Not sure where to go from here.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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"Abstract Algebra - Cyclic groups"

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