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:
$$C_2\oplus C_2\oplus C_2\oplus C_2$$
$$C_4\oplus C_2\oplus C_2$$
$$C_4\oplus C_4$$
$$C_2\oplus C_2\oplus C_2\oplus C_2$$
$$C_4\oplus C_2\oplus C_2$$
$$C_4\oplus C_4$$