A External Direct Sum of Groups Problem

In summary, the conversation is about finding a subgroup of Z_4 \oplus Z_2 that is not of the form H \oplus K, where H is a subgroup of Z_4 and K is a subgroup of Z_2. The conversation discusses the initial confusion and the suggestion to consider a subgroup generated by (2,1) which has order two and is not of the form H+K.
  • #1
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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?
 
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  • #2
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?
 
  • #3
I understand now. Thanks for the tip.
 

Related to A External Direct Sum of Groups Problem

1. What is an external direct sum of groups?

An external direct sum of groups is a mathematical concept that involves combining two or more groups together in a specific way. It is denoted by the symbol ⊕ and is often used to represent the direct product of two or more groups.

2. How is an external direct sum of groups different from an internal direct sum?

An external direct sum of groups involves combining groups that are not necessarily subgroups of each other, while an internal direct sum involves combining subgroups of a larger group. Additionally, the external direct sum is defined using a direct product, while the internal direct sum is defined using a direct sum.

3. What properties does an external direct sum of groups have?

An external direct sum of groups has several properties, including commutativity, associativity, and distributivity. It also has a unique identity element and inverse elements for each group in the sum.

4. How is an external direct sum of groups used in mathematics?

An external direct sum of groups is used in various areas of mathematics, including abstract algebra, group theory, and topology. It is often used to study the structure and properties of groups and to solve problems related to them.

5. Can an external direct sum of groups be infinite?

Yes, an external direct sum of groups can be infinite. In fact, it is often used to combine an infinite number of groups together to create a new group with specific properties. However, it is important to note that the external direct sum of an infinite number of groups may not always be well-defined.

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