List all abelian groups that have order 81 and contain an element of order 27. For each, give the primary decomposition and a specific element having order 27.
I know $81 = 3^{4}$
so the abelian groups are
$\mathbb{Z}_{81}$
$\mathbb{Z}_{27}\times \mathbb{Z}_3$
$\mathbb{Z}_{9}\times \mathbb{Z}_3 \times \mathbb{Z}_3$
$\mathbb{Z}_{3}\times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3$
and I know $3$ in $\mathbb{Z}_{81}$ will have order $27$
but I'm having trouble checking the other 3 groups for an element of order 27