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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?

    Thanks!
     
  2. jcsd
  3. Sep 24, 2014 #2

    Stephen Tashi

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

    mathwonk

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    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:

    http://www.math.uga.edu/~roy/4050sum08.pdf
     
  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

    mathwonk

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    you are welcome. i had fun thinking that stuff through for a summer course in linear algebra, especially the analogies with finite abelian groups.
     
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