Part hasn't registered in group (7)
I believe the answer is:
faction
'group' is the definition.
'part hasn't registered' is the wordplay.
'part' becomes 'fraction' (fraction is a kind of part).
'hasn't' indicates named letters should be taken away.
'registered' becomes 'R' (abbreviation, as in registered trademark).
'fraction' with 'r' removed is 'FACTION'.
'in' acts as a link.
(Other definitions for faction that I've seen before include "A dissenting clique" , "Clique that seeks power often through intrigue" , "group of rebels" , "Rebellious group" , "camp" .)