1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Groups/cyclic groups

  1. Oct 17, 2009 #1
    I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative.


    Also, Consider Zn = {0,1,....,n-1}
    a. show that an element k is a generator of Zn if and only if k and n are relatively prime.

    b. Is every subgroup of Zn cyclic? If so, give a proof. If not, provide an example.
     
  2. jcsd
  3. Oct 18, 2009 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    What does it mean that G is commutative?
    What possible ways to prove commutativity do you know of?

    For the second one, I suggest starting with the "<==" implication (i.e. assume that k and n are relatively prime and show that k generates Zn.
     
  4. Oct 18, 2009 #3


    for the first one, we can show commutativity with a multiplication table. How else?

    for the second one, i want to start with ==> and say that the order of k is n/(m,n). but how can i show it?
     
  5. Oct 19, 2009 #4

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    OK let's take them one at a time.

    The definition of commutativity is that xy = yx for any two elements x and y.
    Can you explicitly show this in the case given?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Groups/cyclic groups
  1. Cyclic Groups (Replies: 6)

  2. Grouping Elements (Replies: 1)

Loading...