Assalamu Alaikum
Salaams initiated from a man to a man are ok.
Salaams initiated from a woman to a woman are ok.
Salaams initiated from a man to his mahram women are ok.
Salaams initiated from a woman to a man or from a man to a woman are disliked for fear of fitnah. However if a man initiates a salaams to an elderly woman, that is permissible as long as there's no temptation.
Quote:
If she is an old woman and he will not be tempted by her, then it is permissible for her to greet the man with salaam and for the man to return her salaams.
If there is a group of women then a man may greet them with salaam, or if there is a group of men, they may greet a woman with salaam, so long as there there is no fear that any of the parties may be tempted.
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Quote:
Al-Nawawi said in his book al-Adhkaar (p. 407):
Our companions said: Women greeting women is like men greeting to men. But when it comes to women greeting men, if the woman is the man's wife, or his concubine, or one of his mahrams, then it is like him speaking to another man; it is mustahabb for either of them to initiate the greeting of salaam and the other is obliged to return the greeting. But if the woman is a stranger (non-mahram), if she is beautiful and there is the fear that he may be tempted by her, then the man should not greet her with salaam, and if he does then it is not permissible for her to reply; she should not initiate the greeting of salaam either, and if she does, she does not deserve a response. If he responds then this is makrooh.
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Quote:
Al-Haafiz said in al-Fath:
Concerning the permissibility of men greeting women with salaam and women greeting men: what is meant by its being permitted is when there is no fear of fitnah.
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You can read more about it in depth
here inshallah.
Allah Knows best.
fi aman Allah
w'salaam