It is not permissible to shake hands with the opposite sex who are not Mahram to you even if they were too old.

Is it allowed for one to shake the hand of an old lady who is not a mahram (she is a relative, but not a mahram)?

Mahram | February 17, 2014 9:48 pm
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Shaykh. I started my PhD in the UK in 2008. At that time I did not know that a woman could not travel without a mahram. Actually, a local shaykh told me that since plane journeys are safe these days it is permissible for a woman to travel on her own.
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I am planning to go for Umrah with my family. My wife’s mother wants to accompany us is it permissible. As per my knowledge women can not go on journey without the mehram(Brother,Father,Husband or son) or any such relation person accompany them?