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

Are All Indecomposable Groups Cyclic?

  1. Sep 23, 2014 #1
    A group is said to be indecomposable if it cannot be written as a product of smaller groups. An example of this is any group of prime order p, which is isomorphic to the group of integers modulo p (with addition as the group operation). Since the integers modulo p is a cyclic group (generated by 1), we have that any indecomposable group of prime order is cyclic. I have two questions:

    Are ALL indecomposable groups cyclic? (N.B. not just those of prime order)

    Are all cyclic groups indecomposable?

  2. jcsd
  3. Sep 24, 2014 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    Look at a list of "simple" groups.
  4. Sep 28, 2014 #3


    User Avatar
    Science Advisor
    Homework Helper

    If you restrict to finite abelian groups, then all indecomposable such groups are cyclic, but not all cyclic groups are indecomposable. All finite abelian groups decompose into a product of not just cyclic factors, but of indecomposable cyclic factors.

    see my notes, especially pages 42-43:

  5. Sep 28, 2014 #4
    Ah, thanks @mathwonk! Your notes turned out to be just the reference I was looking for. I thought the extension to linear mappings was pretty interesting.
  6. Oct 2, 2014 #5


    User Avatar
    Science Advisor
    Homework Helper

    you are welcome. i had fun thinking that stuff through for a summer course in linear algebra, especially the analogies with finite abelian groups.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Indecomposable Groups Cyclic Date
I Lorentz Group Wednesday at 6:29 PM
A Why a Lie Group is closed in GL(n,C)? Feb 12, 2018
A Example of how a rotation matrix preserves symmetry of PDE Feb 10, 2018
A Tensor symmetries and the symmetric groups Feb 9, 2018
Freely Indecomposable Sep 22, 2006