Hw9 Problem 8

### Problem 8

(*) Up to isomorphism, how many Abelian groups of order $16$ have the property that $x + x + x + x = 0$ for all $x$ in the group?

Solution

The given property requires all elements of the group to have order $1$, $2$, or $4$ so that $x + x + x + x = 0$. Then using the Fundamental Theorem of Finite Abelian Groups, the three possibilities we have are

(1)
\begin{align} \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2, \end{align}
(2)
\begin{align} \mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_2, \end{align}

and

(3)
\begin{align} \mathbb{Z}_4 \times \mathbb{Z}_4. \end{align}