Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Determine if a group is cyclic

  1. Sep 17, 2015 #1
    Hello all!

    If I have a group of order 20 that has three elements of order 4, can this group be cyclic? What if it has two elements? I am new to abstract algebra, so please keep that in mind!

    Thanks!
     
  2. jcsd
  3. Sep 17, 2015 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If it's cyclic then it has a generator element g such that ##g^{20}=1## and ##1,g,g^2,...,g^{19}## are all different.

    Let the three elements of order 4 be a, b and c.

    What can we deduce about what powers of g each of those elements could be?
     
  4. Sep 18, 2015 #3
    Do the powers need to divide 20?
     
  5. Sep 18, 2015 #4

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That's a sufficient, but not a necessary condition.

    Think about the* cyclic group of order 20: {1,##g,g^2,...,g^{19}##}. Express the fourth power of each of its elements as ##g^m## where ##m<20##.

    *Note the use of 'the' rather than 'a'. All cyclic groups of order 'n' are isomorphic.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Determine if a group is cyclic
  1. Cyclic Group (Replies: 1)

  2. Cyclic group question (Replies: 2)

Loading...