Please give your comments on the correctness of the following. Thanks.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Z_4 is not a simple group.

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