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Creating noncyclic groups of certain order

  1. Oct 10, 2013 #1

    Entropee

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    How would I construct noncyclic groups of whatever order I want? For example g is order 8.
     
  2. jcsd
  3. Oct 10, 2013 #2

    Office_Shredder

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    If you want an even order the dihedral group is a good example... for odd I don't really know how to do it. Based on the wikipedia article

    http://en.wikipedia.org/wiki/List_of_small_groups

    in which only a single noncyclic odd group is given, Z3xZ3, there apparently aren't many of them.
     
  4. Oct 11, 2013 #3

    lavinia

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    a group of prime order must be cyclic

    if the order is nxm can you think of an example?
     
  5. Oct 11, 2013 #4

    Entropee

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    We'll in some cases I can but for example if the order is 25 I can't think of one that is NON cyclic, otherwise it would be easy.
     
  6. Oct 11, 2013 #5

    pasmith

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    [itex]C_5 \times C_5[/itex] is an abelian group of order 25 in which every element other than the identity is of order 5.

    ([itex]C_n \times C_m[/itex] is not isomorphic to [itex]C_{n+m}[/itex] unless [itex]n[/itex] and [itex]m[/itex] are coprime.)
     
  7. Oct 12, 2013 #6

    lavinia

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    here is a group of order 6.

    b[itex]^{3}[/itex] = a[itex]^{2}[/itex] = id

    aba[itex]^{-1}[/itex] = b[itex]^{2}[/itex]

    Try to show that every group of order 15 is abelian.
     
    Last edited: Oct 12, 2013
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