- #1
peteryellow
- 47
- 0
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.
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.
