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

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The discussion centers on proving that the groups U_{m/n_1}(m) and U_{m/n_k}(m) are normal subgroups of U(m). The user has successfully demonstrated the normality of these subgroups and seeks assistance in proving that U(m) is the direct product of these subgroups and that their intersection is solely the identity element. The notation U(m) refers to the group of units modulo m, which consists of integers that are coprime to m.

PREREQUISITES
  • Understanding of group theory, specifically normal subgroups.
  • Familiarity with the notation and properties of U(m), the group of units modulo m.
  • Knowledge of direct products of groups and their implications.
  • Basic concepts of intersection of groups and identity elements.
NEXT STEPS
  • Research the properties of U(m) and its structure as a group.
  • Study the criteria for normal subgroups and their significance in group theory.
  • Explore the concept of direct products in group theory and how they relate to subgroup structures.
  • Investigate examples of U_{m/n}(m) for various values of m and n to solidify understanding.
USEFUL FOR

Mathematicians, particularly those specializing in abstract algebra, group theorists, and students studying the properties of modular arithmetic and unit groups.

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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