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Cyclic group

  • Thread starter Fairy111
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  • #1
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Homework Statement



How do i go about proving that a group is cyclic?

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
dx
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Start with the definition of a cyclic group, and see if your group satisfies it.
 
  • #3
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The group, G, is a finte group with cardinality p, a prime integer. How should i start off, if i need to prove it's cyclic?
 
  • #4
dx
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Are you familiar with Lagrange's theorem?
 
  • #5
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yes, i know that the only subgroups of G are itself and the subgroup {e} which consists of the neutral element. This is because the only possibilities of the cardinalities of the subgroups are 1 or p.
 
  • #6
dx
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Ok, now pick an element of the group G, say g not equal to 1. What are the possible orders of g?
 
  • #7
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possible orders of g are 1 or p? Since those are the only numbers that divide the prime number p.
 
  • #8
dx
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It cannot be 1 because we assumed g was not equal to the identity. So the order of g must be p, and therefore G = {1 , g, g2, ... , gp-1} which is cyclic.
 
  • #9
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Ok, thanku very much for the help:)
 
  • #10
dx
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No problem.
 

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