Yes, of course. A domain name is not owned after it is registered, but rather leased for a set period of time (usually 1 to 10 years). To avoid having to worry about renewals, most people and businesses set up their domains to auto-renew with their reseller.
If you want to make a website, Google does not provide web hosting. When you purchase web hosting from one of the providers listed below, you will receive a free domain name for a one-year period.
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