Is it permissible to keep silent regarding the crime of adultery practiced by a neighbouring married woman with her husband's friend?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

Adultery is a great evil and a heinous offence, and he who firmly knows about it must forbid it according to his own ability, without falling into slandering, because slandering carries the punishment of flogging. That is, he who accuses someone of adultery and fails to bring four witnesses deserves the corporal punishment prescribed for slandering. Allaah Says (what means): {And those who accuse chaste women and then do not produce four witnesses - lash them with eighty lashes and do not accept from them testimony ever after. And those are the defiantly disobedient, except for those who repent thereafter and reform, for indeed, Allaah is Forgiving and Merciful.}[Quran 24:4-5]

Therefore, start by advising such a woman, reproaching her and threatening her that unless she quits, you will inform her husband of this sinful relation. You may give her this piece of advice directly or by sending her a letter. If she responds to it, then this is the sought objective, and you must keep this a secret and not reveal it to anybody.

Otherwise, inform her husband indirectly, without slandering her. You can say, for example, "Take care of your wife", "Strange people meet your wife" or "Your wife meets other men" It is preferable to send this in an anonymous letter.

To conclude, it is not at all permissible to remain silent regarding such a great evil; it is obligatory to forbid it and try to prevent it. 

Allaah Knows best.