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May 8th, 2019, 06:04 AM  #1 
Member Joined: Apr 2017 From: India Posts: 73 Thanks: 0  Generators of a cyclic group
The number of elements of order 12 in a cyclic group of order 12 is 4 or 3? As I understood, I need to calculate the number of generators of this cyclic group as every generator will have the order 12. To find that I applied the Euler Phi function to find the natural numbers <12 and relatively prime to 12. They come out to be 1,5,7 and 11. That means there are 4 generators and hence the number of elements of order 12 should be 4. But the answer given is 3. Am I wrong somewhere or there is a misprint in my textbook? 
May 8th, 2019, 06:20 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,939 Thanks: 2210 
What is the ISBN of the textbook?

May 8th, 2019, 08:04 AM  #3 
Member Joined: Apr 2017 From: India Posts: 73 Thanks: 0 
I don't know the ISBN, the book is too old. It is given as a multiple choice question with answer marked as 3. However, my conclusion is 4. What's your conclusion? 
May 8th, 2019, 03:53 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
A big part of learning math is developing critical thinking. This includes reading material from trusted sources. This means instead of asking here what the correct answer is, you should just take a cyclic group of order 12 and check since this amounts to about 30 seconds of computation. This should also help reinforce the reason WHY $\phi(n)$ is the number of elements of order $n$ in a cyclic group of order $n$.

May 9th, 2019, 05:36 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,939 Thanks: 2210  

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