Part of Republican inside rebellious group (8)
I believe the answer is:
fraction
'part' is the definition.
(I know that fraction is a type of part)
'republican inside rebellious group' is the wordplay.
'republican' becomes 'R' (abbreviation in US politics).
'inside' is an insertion indicator.
'rebellious group' becomes 'faction' (I've seen this before).
'r' placed inside 'faction' is 'FRACTION'.
'of' acts as a link.
(Other definitions for fraction that I've seen before include "Inca fort (anag)" , "Not a whole number - it's smaller" , "Number of mathematicians" , "Proportional" , "Fragment" .)