Prove that U_{m/n_1} (m) ,U_{m/n_k} (m) are normal subgroups

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In summary, the conversation is discussing a statement asking for a proof that two specific groups, U_{m/n_1} (m) and U_{m/n_k} (m), are normal subgroups. A normal subgroup is a subgroup that is invariant under conjugation by elements of the larger group. The conversation also explains the meaning of a subgroup as a subset of a larger group that forms a group, and the significance of proving normality in understanding the structure and properties of the larger group. This proof may have various applications in mathematics and practical fields.
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vish_maths
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prove that U_{m/n_1} (m) , ... U_{m/n_k} (m) are normal subgroups

In the attached image I have proved that U_{m/n_1} (m) , ... U_{m/n_k} (m) are normal subgroups

But how do i Prove that U(m) = U_{m/n_1} (m) ... U_{m/n_k} (m)?

and that their intersection is identity alone.

Help will be appreciated. Thanks
 

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I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
What is U(m)?
 

FAQ: Prove that U_{m/n_1} (m) ,U_{m/n_k} (m) are normal subgroups

What does "Prove that U_{m/n_1} (m) ,U_{m/n_k} (m) are normal subgroups" mean?

This statement is asking for a proof that two specific groups, U_{m/n_1} (m) and U_{m/n_k} (m), are normal subgroups. A normal subgroup is a subgroup that is invariant under conjugation by elements of the larger group.

What is a subgroup?

A subgroup is a subset of a larger group that itself forms a group. This means that the subgroup contains elements that can be combined using the same operation as the larger group and still produce elements within the subgroup.

What does it mean for a subgroup to be normal?

A normal subgroup is one that is invariant under conjugation by elements of the larger group. In other words, if an element of the larger group is multiplied by an element of the normal subgroup and then multiplied by the inverse of that element, the resulting element is still within the normal subgroup.

Why is it important to prove that U_{m/n_1} (m) ,U_{m/n_k} (m) are normal subgroups?

Proving that these subgroups are normal is important because it helps us understand the structure and properties of the larger group. It also allows us to apply theorems and techniques specific to normal subgroups in further mathematical analysis.

What are some potential applications of this proof?

This proof may have applications in group theory, number theory, and other areas of mathematics. It could also be used in practical applications such as cryptography and computer science.

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