I got a email saying a purchased a game and I didn’t?

If the only indication of the charges is from an email or text, be aware that it may be a phishing attempt to get you to log in to a site that appears to be Apple and divulge your password or credit card information. See: Avoid phishing emails, fake 'virus' alerts, phony support calls, and other scams - Apple Support

and Identify legitimate emails from the App Store or iTunes Store - Apple Support

You can use the following instructions to determine if it is a real charge, what it is for, and to report a problem: Get help with an item purchased from the App Store, iTunes Store, or iBooks Store - Apple Support

It is most likely a phishing attempt. See below for steps to identify bogus messages.

Avoid phishing emails, fake 'virus' alerts, phony support calls, and other scams - Apple Support