Determing the isomorphism class of $U(16)$ and writing it as an internal direct product of cyclic groups.

Question

Determime the isomorphism class of $U(16)$ and write $U(16)$ as an internal direct product of cyclic groups.

This is what I have done so far:

$U(16) = \{1, 3, 5, 7, 9, 11, 13, 15\}$

$|U(16)|=8$

$(element, order) = (1,1), (3,4), (5,4),(7,2),(9,2),(11,4),(13,4),(15,2)$

Need help on how to determine the ismorphism class and how I can use that to write the direct product of cyclic groups.

Answer 1

First, you should write down the possible isomorphism classes of abelian groups of order $8$. The first one, obviously, is the class of $\Bbb Z_8$ (a cyclic group of order $8$). There are two more.

For simplicity's sake, determine the number of elements of order $2$ in each isomorphism class. Then count the number of elements of order $2$ in your group, and see which one it matches.