The carp cooked for a branch of society (7)
I believe the answer is:
chapter
'for a branch of society' is the definition.
'the carp cooked' is the wordplay.
'cooked' is an anagram indicator (letters cooked into a new form).
'the'+'carp'='thecarp'
'thecarp' is an anagram of 'CHAPTER'.
(Other definitions for chapter that I've seen before include "Ecclesiastical assembly or subdivision of a written work" , "one section of book" , "Division of novel - church body" , "Division of book, usually numbered" , "series of events" .)