A External Direct Sum of Groups Problem

[e(ho0n3]

Homework Statement
Find 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 attempt at a solution
I'm guessing I need to find an H \oplus K where either H or K is not a subgroup. But this seems impossible. Obviously (0, 0) will be in H \oplus K so 0 is in H and 0 is in K. If (a, b) and (c, d) are elements of H \oplus K, (a, b) + (c, d) = (a + c, b + d) is in H \oplus K 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?

 

[Dick]

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?

 

[e(ho0n3]

I understand now. Thanks for the tip.