Difference Between Debian Standard and GNOME – Linux Desktop Comparison

aptitude purge `dpkg --get-selections | grep gnome | cut -f 1`
aptitude -f install
aptitude purge `dpkg --get-selections | grep deinstall | cut -f 1`
aptitude -f install

@rob is right. GNOME is technically an official GNU project. However, there is a lot of interesting history.

Let's roll back the clock

It's 1996. There are no desktop environments. Users and sysadmins assemble environments from a hodge-podge of programs. Different window managers, different applications, maybe a dock. There are two major toolkits on the market: Qt and GTK+. Qt had been around for a while, and was a commercial product of a company called Trolltech. GTK+ had also been around for a fair while. It was loosely associated with the FSF, since it was originally written for use in the GIMP. There were more toolkits, like (for example) Motif, but for the purposes of this discussion, we don't care about them.

The Kool Desktop Environment, also known as KDE, was created in October of that year in response to the fact that there was no unified desktop environment for UNIX systems. (The KDE project quickly dropped "Kool" in favor of just an undefined "K". It was clearly a good choice.) The creator of KDE, Matthias Ettrich, chose to use Qt for his new desktop. This was a major problem for the free software community. It meant that in order to use the awesome, free desktop that Matthias had created, they would have to install proprietary software - Qt.

What to do?

The FSF responded with not one but two projects, both working in parallel just in case one didn't pan out. The first was a project called Harmony. Harmony was intended to be an LGPL-licensed, API-compatible free software clone of Qt. The idea was that the community would keep KDE, simply replacing the proprietary bit.

The Harmony project never really worked out. Development went on for about 4 years before Qt was relicensed in 2000 to be fully free software (as defined by the FSF), thus eliminating the original motivation for Harmony. Due to both the relicensing and the success of the second project, Harmony was abandoned.

I bet you've guessed what the second project was by now. It was GNOME.

Tying it all together

I've given the history above. Now let's tie it all together in a nice knot.

So, to answer your question: yes, there is a relationship between GNU and GNOME. GNOME is the official desktop environment of the GNU project and is therefore an official GNU project and a part of the GNU operating system. Historically, it was created by GNU in response to KDE's dependence on Qt. In fact, the G in GNOME stands for GNU. The full acronym expands to GNU Network Object Model Environment - this refers to a technology that was planned but never implemented, as the project decided that it "didn't fit with the core GNOME vision".

That being said, GNOME is a huge project now. GTK+ is maintained by the GNOME people nowadays, for example, instead of being an independent project. It is safe to say that GNOME as an entity is independent of GNU, even though they are historically and technically related. GNOME has its own infrastructure; its own community; its own governance processes.

As a side note, this is also why GNOME and KDE are (friendly) rivals nowadays. It is because back in 1996, when KDE was founded, GNOME was created with the express purpose of directly competing with KDE. And that rivalry has persisted all the way up until the present.