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Z_4 is not a simple group.

  1. Aug 17, 2008 #1
    Please give your comments on the correctness of the following. Thanks.

    My question is :
    Give an example of a Z-module of finite length not being semisimple.


    I want to show that Z_4 where Z = integers is not simple.

    2Z_4 is a subgroup of order 2 which is normal subgroup, hence Z_4 is not simple.

    But how can I argument that this module is not semisimple? Z_4 can not be written as
    a direct sum of simple subgroups because the only subgroups of Z_4 are (0), 2Z_4 and Z_4.
    So we can not write Z_4 as a direct sum of proper subgroups.

    Is it correct. And this module has finite length since 0/2Z_4 =0 and 2Z_4/Z_4 =0.
     
  2. jcsd
  3. Aug 18, 2008 #2

    morphism

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    Looks OK to me.
     
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