Question Summary:
is this allow for a muslim man or woman to greet the islamic greeting asalamu alaikum to each other

Question Detail:

is this allow for a muslim man or woman to greet the islamic greeting asalamu alaikum to each other

Answer :

In the name of Allah, Most Gracious, Most Merciful
Assalaamu `alaykum waRahmatullahi Wabarakatuh
Due to the risk of Fitna (corruption and immorality), it is inappropriate for non-Mahram men and women to greet each other with Assalaamu Alaykum or any other greeting. However, if a man is greeted by an elderly woman or he greets the elderly woman it is permissible for the one greeted to reply. Conversely, if a young woman greets a man or a man greets a young woman, the one greeted should not reply verbally; rather, he or she should reply within his or her heart.  (Fatawa Rahimiya, 10/126, Isha’at)
وإذا سلمت المرأة الأجنبية على رجل إن كانت عجوزا رد الرجل عليها السلام بلسانه بصوت تسمع وإن كانت شابة رد عليها في نفسه والرجل إذا سلم على امرأة أجنبية فالجواب فيه على العكس
الفتاوى العالمكيرية، 5/326، رشيديه) )
 
And Allah knows best
Wassalamu Alaikum
Ml. Sajid bin Shabbir,
Student Darul Iftaa
Checked and Approved by:
Mufti Ebrahim Desai
Darul Iftaa, Madrassah In'aamiyyah