A External Direct Sum of Groups Problem

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SUMMARY

The discussion centers on identifying a subgroup of the direct sum of groups Z_4 ⊕ Z_2 that cannot be expressed as H ⊕ K, where H is a subgroup of Z_4 and K is a subgroup of Z_2. The user initially struggles with the concept, mistakenly assuming that all subgroups must conform to the H ⊕ K structure. However, they realize that the subgroup generated by (2,1) has order two and prompts further investigation into which subgroups of the form H ⊕ K also have order two. This leads to a clearer understanding of subgroup generation in the context of direct sums.

PREREQUISITES
  • Understanding of group theory concepts, specifically direct sums of groups.
  • Familiarity with the structure of finite groups, particularly Z_4 and Z_2.
  • Knowledge of subgroup generation and properties of subgroups.
  • Basic algebraic operations within groups, such as addition of group elements.
NEXT STEPS
  • Research the properties of direct sums in group theory.
  • Explore subgroup generation techniques in finite groups.
  • Study examples of subgroups in Z_4 and Z_2 to identify their structures.
  • Learn about the classification of finite abelian groups and their subgroup lattices.
USEFUL FOR

Students of abstract algebra, mathematicians focusing on group theory, and educators teaching concepts related to finite groups and subgroup structures.

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Homework Statement
Find a subgroup of [itex]Z_4 \oplus Z_2[/itex] that is not of the form [itex]H \oplus K[/itex] where H is a subgroup of [itex]Z_4[/itex] and K is a subgroup of [itex]Z_2[/itex].

The attempt at a solution
I'm guessing I need to find an [itex]H \oplus K[/itex] where either H or K is not a subgroup. But this seems impossible. Obviously (0, 0) will be in [itex]H \oplus K[/itex] so 0 is in H and 0 is in K. If (a, b) and (c, d) are elements of [itex]H \oplus K[/itex], (a, b) + (c, d) = (a + c, b + d) is in [itex]H \oplus K[/itex] so a, c, and a + b must be in H and b, d, and b + d must be in K. Since these are finite groups, H and K must be subgroups by closure. What's going on?
 
Physics news on Phys.org
You want to find a subgroup that's NOT of the form H+K. Don't assume it's of the form H+K to begin with. Consider the subgroup generated by (2,1)? It has order two. What subgroups of the form H+K have order two. Is this one of them?
 
I understand now. Thanks for the tip.
 

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