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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?