Z_4 is not a Semisimple Z-Module of Finite Length: Explanation and Example

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In summary, the conversation discusses giving an example of a Z-module of finite length that is not semisimple. It is argued that Z_4, where Z represents integers, is not simple because it can be written as a direct sum of subgroups. However, it is also shown that Z_4 cannot be written as a direct sum of proper subgroups, indicating that it is not semisimple. The correctness of this argument is confirmed and it is noted that this module has finite length.
  • #1
peteryellow
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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.
 
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  • #2
Looks OK to me.
 

FAQ: Z_4 is not a Semisimple Z-Module of Finite Length: Explanation and Example

What does it mean for a group to be simple?

A simple group is a group that has no proper nontrivial normal subgroups. This means that the only subgroups of a simple group are the group itself and the trivial subgroup containing only the identity element.

Why is Z_4 not a simple group?

Z_4, also known as the cyclic group of order 4, is not a simple group because it has a nontrivial normal subgroup. In fact, Z_4 has two proper nontrivial normal subgroups: {0, 2} and {0, 1, 2, 3}.

What are the elements of Z_4?

The elements of Z_4 are 0, 1, 2, and 3, where the operation is addition modulo 4. This means that 0 is the identity element, and every element is its own inverse.

How is the simplicity of a group related to its structure?

The simplicity of a group is closely related to its structure. A simple group cannot be decomposed into smaller groups, which means that its structure is "simple" and cannot be broken down into simpler components.

What are some examples of simple groups?

Some examples of simple groups include the alternating groups, the symmetric groups, and the sporadic simple groups. The simple groups also play a crucial role in the classification of finite simple groups, which is one of the most significant achievements in group theory.

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