# A External Direct Sum of Groups Problem

1. Jun 10, 2008

### e(ho0n3

The problem statement, all variables and given/known data
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?

2. Jun 10, 2008

### 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?

3. Jun 11, 2008

### e(ho0n3

I understand now. Thanks for the tip.