I know that you should say Salaam to every brother who passes by whether you know him or not. Should a man passing a woman give Salaam to her? I have heard a Hadith that when the Messenger of Allah passed a sister, he would say greetings. Is this true?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

At-Tirmithi reported from Asmaa’ bint Yazeed that she said: “The Prophet passed by us (a group of women) and greeted us."

So, saying "Assalamu Alaykum" to a woman is lawful provided there is no fear of temptation. An-Nawawi reported from Imam Abu Sayyid al-Mutawally that he said: “If the woman is a non-Mahram to the man and if she is beautiful and if there is fear of temptation, then the man should not greet that woman and if he greets her she should not reply. Now if such a woman greets the man first, he should not reply. If he replies to her greeting, that is dislikable for him. But if the woman is aged hence not tempting, then it is lawful for her to start her greeting to the man and that man should reply to her. It is established that the man greets the group of women, and the group of men greet the individual woman as long as there is no fear of temptation.”

Al-Bukhaari entitled a chapter in his famous book with this title: “Chapter of men greeting women and vice versa.” Ibn Hajar commented on this saying: “The lawfulness of men greeting women and vice versa is limited to the situation where there is no fear of temptation. Al-Halimi said: 'there was no fear of temptation, as far as the Prophet is concerned as he was infallible. So, anyone who is confident that he would not be tempted can greet; otherwise, he should remain silent.’”

For more benefit on permissibility of greeting non-Mahram women if there is no fear of temptation, please refer to Fataawa 347464, 387343, and 82299.

Allah knows best.