Is UNIX-operating system a free software?

Currently, The Open Group owns the "Unix" trademark.

No, Unix branded operating systems are proprietary and copyrighted. They are not free or open systems software.

From its start until 1993, Unix distributions included all their source code.

Programmers have made many free and open Unix-like operating systems, including Linux, NetBSD, FreeBSD, OpenBSD, Dragonfly BSD, Mach, OpenSolaris, etc.