not "conduction", but ___?

System X is designed as an environment where Conversation Y can be conducted by the members of Group Z.

Equivalently:

System X is designed to facilitate the __? __ of Conversation Y by the members of Group Z.

Conduction doesn't feel right.

Solution 1:

The noun is "conduct". You could say, "System X is designed to facilitate the conduct of conversation Y."

Note it is pronounced differently from the verb. The verb "conduct" is pronounced with the accent on the second syllable; the noun is pronounced with the accent of the first syllable.

In any case, it's unnecessarily wordy. You could just leave it out. "System X is designed to facilitate conversation Y." But if you want to make it clear that you are not talking about the conversation itself, but the action of having the conversation, you could use it. I'm not sure what the significance of such a distinction would be, but it might make sense in context.

"Conduct of" is sometimes used to emphasize the way something is done, rather than the thing itself. Like, "The use of poison gas changed the conduct of the war", or "The new banking regulations are an attempt to influence the conduct of banking in America."

Solution 2:

I think 'Conversation' is the word you need. The sentence doesn't need the word; just say

System X is designed to facilitate Conversation Y by the members of Group Z.

Solution 3:

Progress, defined by Cambridge Dictionaries Online

movement toward an improved or more developed state, or to a forward position:

The talks failed to make any progress toward a settlement

In the OP's example:

System X is designed to facilitate the progress of Conversation Y by the members of Group Z

The meaning of facilitating the progress is that System X is pulling the conversation back on track, whenever it gets too far off topic for too long. Of course, any conversation will veer off topic, and this can actually be productive as participants make unexpected connections, but at some point, it hampers progress.