## Group Theory problem

### Group Theory problem

Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G?

### Re: Group Theory problem

Every finite abelian group can be expressed as as direct sum of cyclic groups. Since x+x+x+x=0 for each x, the order of each of these cyclic groups is 2 or 4. We have 3 possibilities:

$$C_2\oplus C_2\oplus C_2\oplus C_2$$

$$C_4\oplus C_2\oplus C_2$$

$$C_4\oplus C_4$$

$$C_2\oplus C_2\oplus C_2\oplus C_2$$

$$C_4\oplus C_2\oplus C_2$$

$$C_4\oplus C_4$$