"Oni nie mają kota."
Translation:They do not have a cat.
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Yes. "kawa" is feminine singular, and Genitive singular of feminine nouns is almost always identical to Nominative/Accusative plural. That's quite confusing for learners at first, although after some time it can become helpful - one form less to remember.
"koty" would be the Genitive if the basic form was "kota". But it's not. Indeed "koty" is plural (Nom/Acc) and Genitive plural is "kotów".