The most common and most likely answer to this crossword clue is the 7 letter word MAFIOSI
I've seen this crossword clue in The Daily Telegraph
M
A
F
I
O
S
I
More clues leading to the result MAFIOSI
- they are criminal
- Criminal? If I am, so are they
- Aims of criminal joining Italy’s No 1 illegal gang
- Members of Italian society breaking the rules
- Aims of criminal, one or several
- Gangsters subverting aims of independence
- As if I’m bothered about old criminals
- If I am so bad, they must be too!
- Sham aims of one providing hoods
- Gangsters here? If so I am off!
Buddy explains!

The straight part
We know that "Members of Italian society breaking the rules" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case members of italian society breaking the rules = mafiosi.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does MAFIOSI mean?
MAFIOSI noun- A member of the Mafia.