In spite of the new friends, she is often homesick.
Now, as far the genitive is concerned, you could have, for example:
Das Messer des Mannes - The knife of the man.
Der Wagen der Frau - The car of the woman.
Der Zahn des Kindes - The tooth of the child.
Or, in the plural:
Das Messer der Männer - The knife of the men.
Der Wagen der Frauen - The car of the women.
Der Zahn der Kinder - The tooth of the children.
Here are some examples with trotz:
Trotz des Wetters fahren wir heute nach Hause - In spite of the weather, we're driving home today.
Trotz meines Versuchs - Despite my attempt.
... trotz zahlreicher Proteste - ... notwithstanding numerous protests.
Trotz intensiver Suche - Despite an intensive search.
Trotz aller Bemühungen - Despite all efforts.
Trotz + des... means that what follows is a masculine or neuter noun in the singular.
Trotz + der... could indicate either a feminine noun in the singular, or anything in the plural.
TL;DR: take a look at the following table https://deutsch.lingolia.com/en/grammar/declension/genitive