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!

Homework Help: Group is a union of proper subgroups iff. it is non-cyclic

  1. Dec 8, 2016 #1
    1. The problem statement, all variables and given/known data

    Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic.

    2. Relevant equations


    3. The attempt at a solution

    " => "
    If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one element of G that is not in that particular subgroup. But then we know that none of the subgroups can represent all the elements of G. Therefore, G is not cyclic.

    " <= "
    If the group is not cyclic, then no element a in G generates G. That means that G is the union of all the <a> subgroups for all a in G.

    Is this correct?
  2. jcsd
  3. Dec 8, 2016 #2


    User Avatar
    2017 Award

    Staff: Mentor

    Looks fine to me.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted