Main Entry: faction
Pronunciation: \ˈfak-shən\
Function: noun
Etymology: Middle French & Latin; Middle French, from Latin faction-, factio act of making, faction — more at fashion
Date: 1509
1 : a party or group (as within a government) that is often contentious or self-seeking : clique
2 : party spirit especially when marked by dissension
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Technically faction probably isn't the right word to use to start with, however:
All people using a Mercenary ID are part of the Mercenary faction wherever they live. Mercs in Liberty are part of the Merc faction within Liberty. Groups within that faction include factions of the Merc faction (MM for example). The same goes for every ID in game just like Strom already pointed out.
For example:
The Bounty Hunters Guild is, in itself, a faction. The BHG| is a faction within the Bounty Hunters Guild.
The Hogosha, a faction within Kusari. The Black Dragon Society is a faction within the Hogosha.
The Liberty Rogues, a faction in Liberty. The LR- is a faction within the Liberty Rogues.
The Outcasts, a faction in the Borderworlds. The RoS is a faction of the Outcasts.
The Liberty Navy is a faction of the Liberty government. The [LN] is a faction of the Liberty Navy.
As it stands now no official faction represents their entire faction as a whole (this is how indies exist), so each official faction is a faction of the larger faction. If I lost you, sorry. Use a dictionary and figure it out for yourself.